:: PARTIT_2 semantic presentation

definition
let X be set , Y be set ;
let R be Relation of X,Y, S be Relation of X,Y;
redefine pred c= as c3 c= c4 means :: PARTIT_2:def 1
for x being Element of X
for y being Element of Y st [x,y] in R holds
[x,y] in S;
compatibility
( R c= S iff for x being Element of X
for y being Element of Y st [x,y] in R holds
[x,y] in S )
proof end;
end;

:: deftheorem Def1 defines c= PARTIT_2:def 1 :
for X, Y being set
for R, S being Relation of X,Y holds
( R c= S iff for x being Element of X
for y being Element of Y st [x,y] in R holds
[x,y] in S );

definition
let Y be non empty set ;
let G be non empty Subset of (PARTITIONS Y);
redefine mode Element as Element of c2 -> a_partition of a1;
coherence
for b1 being Element of G holds b1 is a_partition of Y
proof end;
end;

theorem Th1: :: PARTIT_2:1
for Y being non empty set holds '/\' ({} (PARTITIONS Y)) = %O Y
proof end;

theorem Th2: :: PARTIT_2:2
for Y being non empty set
for R, S being Equivalence_Relation of Y holds R \/ S c= R * S
proof end;

theorem Th3: :: PARTIT_2:3
for Y being non empty set
for R being Relation of Y holds R c= nabla Y
proof end;

theorem Th4: :: PARTIT_2:4
for Y being non empty set
for R being Equivalence_Relation of Y holds
( (nabla Y) * R = nabla Y & R * (nabla Y) = nabla Y )
proof end;

theorem Th5: :: PARTIT_2:5
for Y being non empty set
for P being a_partition of Y
for x, y being Element of Y holds
( [x,y] in ERl P iff x in EqClass y,P )
proof end;

theorem Th6: :: PARTIT_2:6
for Y being non empty set
for P, Q, R being a_partition of Y st ERl R = (ERl P) * (ERl Q) holds
for x, y being Element of Y holds
( x in EqClass y,R iff ex z being Element of Y st
( x in EqClass z,P & z in EqClass y,Q ) )
proof end;

theorem Th7: :: PARTIT_2:7
for R, S being Relation
for Y being set st R is_reflexive_in Y & S is_reflexive_in Y holds
R * S is_reflexive_in Y
proof end;

theorem Th8: :: PARTIT_2:8
for R being Relation
for Y being set st R is_reflexive_in Y holds
Y c= field R
proof end;

theorem Th9: :: PARTIT_2:9
for Y being set
for R being Relation of Y st R is_reflexive_in Y holds
Y = field R
proof end;

theorem Th10: :: PARTIT_2:10
for Y being non empty set
for R, S being Equivalence_Relation of Y st R * S = S * R holds
R * S is Equivalence_Relation of Y
proof end;

theorem Th11: :: PARTIT_2:11
for Y being non empty set
for a, b being Element of Funcs Y,BOOLEAN st a '<' b holds
'not' b '<' 'not' a
proof end;

theorem Th12: :: PARTIT_2:12
canceled;

theorem Th13: :: PARTIT_2:13
for Y being non empty set
for a, b being Element of Funcs Y,BOOLEAN
for G being Subset of (PARTITIONS Y)
for P being a_partition of Y st a '<' b holds
All a,P,G '<' All b,P,G
proof end;

theorem Th14: :: PARTIT_2:14
canceled;

theorem Th15: :: PARTIT_2:15
for Y being non empty set
for a, b being Element of Funcs Y,BOOLEAN
for G being Subset of (PARTITIONS Y)
for P being a_partition of Y st a '<' b holds
Ex a,P,G '<' Ex b,P,G
proof end;

Lemma51: for Y being non empty set
for G being Subset of (PARTITIONS Y) st G is independent holds
for P, Q being Subset of (PARTITIONS Y) st P c= G & Q c= G holds
(ERl ('/\' P)) * (ERl ('/\' Q)) c= (ERl ('/\' Q)) * (ERl ('/\' P))
proof end;

theorem Th16: :: PARTIT_2:16
for Y being non empty set
for G being Subset of (PARTITIONS Y) st G is independent holds
for P, Q being Subset of (PARTITIONS Y) st P c= G & Q c= G holds
(ERl ('/\' P)) * (ERl ('/\' Q)) = (ERl ('/\' Q)) * (ERl ('/\' P))
proof end;

theorem Th17: :: PARTIT_2:17
for Y being non empty set
for a being Element of Funcs Y,BOOLEAN
for G being Subset of (PARTITIONS Y)
for P, Q being a_partition of Y st G is independent holds
All (All a,P,G),Q,G = All (All a,Q,G),P,G
proof end;

theorem Th18: :: PARTIT_2:18
for Y being non empty set
for a being Element of Funcs Y,BOOLEAN
for G being Subset of (PARTITIONS Y)
for P, Q being a_partition of Y st G is independent holds
Ex (Ex a,P,G),Q,G = Ex (Ex a,Q,G),P,G
proof end;

theorem Th19: :: PARTIT_2:19
for Y being non empty set
for a being Element of Funcs Y,BOOLEAN
for G being Subset of (PARTITIONS Y)
for P, Q being a_partition of Y st G is independent holds
Ex (All a,P,G),Q,G '<' All (Ex a,Q,G),P,G
proof end;