:: BVFUNC_9 semantic presentation
Lemma19:
for Y being non empty set
for a, b being Element of Funcs Y,BOOLEAN holds a '&' b '<' a
Lemma21:
for Y being non empty set
for a, b, c being Element of Funcs Y,BOOLEAN holds
( (a '&' b) '&' c '<' a & (a '&' b) '&' c '<' b )
Lemma24:
for Y being non empty set
for a, b, c, d being Element of Funcs Y,BOOLEAN holds
( ((a '&' b) '&' c) '&' d '<' a & ((a '&' b) '&' c) '&' d '<' b )
Lemma25:
for Y being non empty set
for a, b, c, d, e being Element of Funcs Y,BOOLEAN holds
( (((a '&' b) '&' c) '&' d) '&' e '<' a & (((a '&' b) '&' c) '&' d) '&' e '<' b )
Lemma26:
for Y being non empty set
for a, b, c, d, e, f being Element of Funcs Y,BOOLEAN holds
( ((((a '&' b) '&' c) '&' d) '&' e) '&' f '<' a & ((((a '&' b) '&' c) '&' d) '&' e) '&' f '<' b )
Lemma27:
for Y being non empty set
for a, b, c, d, e, f, g being Element of Funcs Y,BOOLEAN holds
( (((((a '&' b) '&' c) '&' d) '&' e) '&' f) '&' g '<' a & (((((a '&' b) '&' c) '&' d) '&' e) '&' f) '&' g '<' b )
Lemma28:
for Y being non empty set
for a, b, c, d, e, f, g being Element of Funcs Y,BOOLEAN holds
( ((((((a '&' b) '&' c) '&' d) '&' e) '&' f) '&' g) 'imp' a = I_el Y & ((((((a '&' b) '&' c) '&' d) '&' e) '&' f) '&' g) 'imp' b = I_el Y & ((((((a '&' b) '&' c) '&' d) '&' e) '&' f) '&' g) 'imp' c = I_el Y & ((((((a '&' b) '&' c) '&' d) '&' e) '&' f) '&' g) 'imp' d = I_el Y & ((((((a '&' b) '&' c) '&' d) '&' e) '&' f) '&' g) 'imp' e = I_el Y & ((((((a '&' b) '&' c) '&' d) '&' e) '&' f) '&' g) 'imp' f = I_el Y & ((((((a '&' b) '&' c) '&' d) '&' e) '&' f) '&' g) 'imp' g = I_el Y )
theorem Th1: :: BVFUNC_9:1
theorem Th2: :: BVFUNC_9:2
theorem Th3: :: BVFUNC_9:3
theorem Th4: :: BVFUNC_9:4
theorem Th5: :: BVFUNC_9:5
theorem Th6: :: BVFUNC_9:6
theorem Th7: :: BVFUNC_9:7
theorem Th8: :: BVFUNC_9:8
theorem Th9: :: BVFUNC_9:9
theorem Th10: :: BVFUNC_9:10
theorem Th11: :: BVFUNC_9:11
theorem Th12: :: BVFUNC_9:12
theorem Th13: :: BVFUNC_9:13
theorem Th14: :: BVFUNC_9:14
theorem Th15: :: BVFUNC_9:15
theorem Th16: :: BVFUNC_9:16
theorem Th17: :: BVFUNC_9:17
theorem Th18: :: BVFUNC_9:18
theorem Th19: :: BVFUNC_9:19
theorem Th20: :: BVFUNC_9:20
theorem Th21: :: BVFUNC_9:21
theorem Th22: :: BVFUNC_9:22
theorem Th23: :: BVFUNC_9:23
theorem Th24: :: BVFUNC_9:24
Lemma51:
for Y being non empty set
for a1, b1, c1, a2, b2, c2 being Element of Funcs Y,BOOLEAN holds (((((((a1 'imp' a2) '&' (b1 'imp' b2)) '&' (c1 'imp' c2)) '&' ((a1 'or' b1) 'or' c1)) '&' ('not' (a2 '&' b2))) '&' ('not' (a2 '&' c2))) '&' ('not' (b2 '&' c2))) 'imp' (((((a1 'imp' a2) '&' (b1 'imp' b2)) '&' ((a1 'or' b1) 'or' c1)) '&' ('not' (c2 '&' a2))) '&' ('not' (c2 '&' b2))) = I_el Y
Lemma52:
for Y being non empty set
for a1, b1, c1, a2, b2, c2 being Element of Funcs Y,BOOLEAN holds (((((((a1 'imp' a2) '&' (b1 'imp' b2)) '&' (c1 'imp' c2)) '&' ((a1 'or' b1) 'or' c1)) '&' ('not' (a2 '&' b2))) '&' ('not' (a2 '&' c2))) '&' ('not' (b2 '&' c2))) 'imp' (((((a1 'imp' a2) '&' (c1 'imp' c2)) '&' ((a1 'or' b1) 'or' c1)) '&' ('not' (b2 '&' a2))) '&' ('not' (b2 '&' c2))) = I_el Y
Lemma53:
for Y being non empty set
for a1, b1, c1, a2, b2, c2 being Element of Funcs Y,BOOLEAN holds (((((((a1 'imp' a2) '&' (b1 'imp' b2)) '&' (c1 'imp' c2)) '&' ((a1 'or' b1) 'or' c1)) '&' ('not' (a2 '&' b2))) '&' ('not' (a2 '&' c2))) '&' ('not' (b2 '&' c2))) 'imp' (((((b1 'imp' b2) '&' (c1 'imp' c2)) '&' ((a1 'or' b1) 'or' c1)) '&' ('not' (a2 '&' b2))) '&' ('not' (a2 '&' c2))) = I_el Y
theorem Th25: :: BVFUNC_9:25
theorem Th26: :: BVFUNC_9:26
theorem Th27: :: BVFUNC_9:27
theorem Th28: :: BVFUNC_9:28
theorem Th29: :: BVFUNC_9:29
theorem Th30: :: BVFUNC_9:30
theorem Th31: :: BVFUNC_9:31
theorem Th32: :: BVFUNC_9:32
theorem Th33: :: BVFUNC_9:33
theorem Th34: :: BVFUNC_9:34
theorem Th35: :: BVFUNC_9:35
theorem Th36: :: BVFUNC_9:36
theorem Th37: :: BVFUNC_9:37
theorem Th38: :: BVFUNC_9:38
theorem Th39: :: BVFUNC_9:39
theorem Th40: :: BVFUNC_9:40