let x, y, c be set ; :: thesis: ( x <> [<*y,c*>,'&'] & y <> [<*c,x*>,'&'] & c <> [<*x,y*>,'&'] implies InputVertices (MajorityIStr (x,y,c)) = {x,y,c} )
assume that
A1: x <> [<*y,c*>,'&'] and
A2: y <> [<*c,x*>,'&'] and
A3: c <> [<*x,y*>,'&'] ; :: thesis: InputVertices (MajorityIStr (x,y,c)) = {x,y,c}
A4: 1GateCircStr (<*x,y*>,'&') tolerates 1GateCircStr (<*y,c*>,'&') by CIRCCOMB:55;
A5: y in {1,y} by TARSKI:def 2;
A6: y in {2,y} by TARSKI:def 2;
A7: {1,y} in [1,y] by TARSKI:def 2;
A8: {2,y} in [2,y] by TARSKI:def 2;
<*y,c*> = <*y*> ^ <*c*> by FINSEQ_1:def 9;
then A9: <*y*> c= <*y,c*> by FINSEQ_6:12;
<*y*> = {[1,y]} by FINSEQ_1:def 5;
then A10: [1,y] in <*y*> by TARSKI:def 1;
A11: <*y,c*> in {<*y,c*>} by TARSKI:def 1;
A12: {<*y,c*>} in [<*y,c*>,'&'] by TARSKI:def 2;
then A13: y <> [<*y,c*>,'&'] by A5, A7, A9, A10, A11, ORDINAL1:5;
A14: c in {2,c} by TARSKI:def 2;
A15: {2,c} in [2,c] by TARSKI:def 2;
dom <*y,c*> = Seg 2 by FINSEQ_1:110;
then A18: 2 in dom <*y,c*> by FINSEQ_1:3;
<*y,c*> . 2 = c by FINSEQ_1:61;
then [2,c] in <*y,c*> by A18, FUNCT_1:8;
then A19: c <> [<*y,c*>,'&'] by A11, A12, A14, A15, ORDINAL1:5;
dom <*x,y*> = Seg 2 by FINSEQ_1:110;
then A22: 2 in dom <*x,y*> by FINSEQ_1:3;
<*x,y*> . 2 = y by FINSEQ_1:61;
then A23: [2,y] in <*x,y*> by A22, FUNCT_1:8;
A24: <*x,y*> in {<*x,y*>} by TARSKI:def 1;
{<*x,y*>} in [<*x,y*>,'&'] by TARSKI:def 2;
then y <> [<*x,y*>,'&'] by A6, A8, A23, A24, ORDINAL1:5;
then A25: not [<*x,y*>,'&'] in {y,c} by A3, TARSKI:def 2;
A26: x in {1,x} by TARSKI:def 2;
A27: {1,x} in [1,x] by TARSKI:def 2;
<*x,y*> = <*x*> ^ <*y*> by FINSEQ_1:def 9;
then A28: <*x*> c= <*x,y*> by FINSEQ_6:12;
<*x*> = {[1,x]} by FINSEQ_1:def 5;
then A29: [1,x] in <*x*> by TARSKI:def 1;
A30: <*x,y*> in {<*x,y*>} by TARSKI:def 1;
{<*x,y*>} in [<*x,y*>,'&'] by TARSKI:def 2;
then A31: x <> [<*x,y*>,'&'] by A26, A27, A28, A29, A30, ORDINAL1:5;
A32: not c in {[<*x,y*>,'&'],[<*y,c*>,'&']} by A3, A19, TARSKI:def 2;
A33: not x in {[<*x,y*>,'&'],[<*y,c*>,'&']} by A1, A31, TARSKI:def 2;
A34: x in {2,x} by TARSKI:def 2;
A35: {2,x} in [2,x] by TARSKI:def 2;
dom <*c,x*> = Seg 2 by FINSEQ_1:110;
then A38: 2 in dom <*c,x*> by FINSEQ_1:3;
<*c,x*> . 2 = x by FINSEQ_1:61;
then A39: [2,x] in <*c,x*> by A38, FUNCT_1:8;
A40: <*c,x*> in {<*c,x*>} by TARSKI:def 1;
{<*c,x*>} in [<*c,x*>,'&'] by TARSKI:def 2;
then A41: x <> [<*c,x*>,'&'] by A34, A35, A39, A40, ORDINAL1:5;
A42: c in {1,c} by TARSKI:def 2;
A43: {1,c} in [1,c] by TARSKI:def 2;
<*c,x*> = <*c*> ^ <*x*> by FINSEQ_1:def 9;
then A44: <*c*> c= <*c,x*> by FINSEQ_6:12;
<*c*> = {[1,c]} by FINSEQ_1:def 5;
then A45: [1,c] in <*c*> by TARSKI:def 1;
A46: <*c,x*> in {<*c,x*>} by TARSKI:def 1;
{<*c,x*>} in [<*c,x*>,'&'] by TARSKI:def 2;
then c <> [<*c,x*>,'&'] by A42, A43, A44, A45, A46, ORDINAL1:5;
then A47: not [<*c,x*>,'&'] in {x,y,c} by A2, A41, ENUMSET1:def 1;
InputVertices (MajorityIStr (x,y,c)) = ((InputVertices ((1GateCircStr (<*x,y*>,'&')) +* (1GateCircStr (<*y,c*>,'&')))) \ (InnerVertices (1GateCircStr (<*c,x*>,'&')))) \/ ((InputVertices (1GateCircStr (<*c,x*>,'&'))) \ (InnerVertices ((1GateCircStr (<*x,y*>,'&')) +* (1GateCircStr (<*y,c*>,'&'))))) by CIRCCMB2:6, CIRCCOMB:55
.= ((((InputVertices (1GateCircStr (<*x,y*>,'&'))) \ (InnerVertices (1GateCircStr (<*y,c*>,'&')))) \/ ((InputVertices (1GateCircStr (<*y,c*>,'&'))) \ (InnerVertices (1GateCircStr (<*x,y*>,'&'))))) \ (InnerVertices (1GateCircStr (<*c,x*>,'&')))) \/ ((InputVertices (1GateCircStr (<*c,x*>,'&'))) \ (InnerVertices ((1GateCircStr (<*x,y*>,'&')) +* (1GateCircStr (<*y,c*>,'&'))))) by CIRCCMB2:6, CIRCCOMB:55
.= ((((InputVertices (1GateCircStr (<*x,y*>,'&'))) \ (InnerVertices (1GateCircStr (<*y,c*>,'&')))) \/ ((InputVertices (1GateCircStr (<*y,c*>,'&'))) \ (InnerVertices (1GateCircStr (<*x,y*>,'&'))))) \ (InnerVertices (1GateCircStr (<*c,x*>,'&')))) \/ ((InputVertices (1GateCircStr (<*c,x*>,'&'))) \ ((InnerVertices (1GateCircStr (<*x,y*>,'&'))) \/ (InnerVertices (1GateCircStr (<*y,c*>,'&'))))) by A4, CIRCCOMB:15
.= ((((InputVertices (1GateCircStr (<*x,y*>,'&'))) \ {[<*y,c*>,'&']}) \/ ((InputVertices (1GateCircStr (<*y,c*>,'&'))) \ (InnerVertices (1GateCircStr (<*x,y*>,'&'))))) \ (InnerVertices (1GateCircStr (<*c,x*>,'&')))) \/ ((InputVertices (1GateCircStr (<*c,x*>,'&'))) \ ((InnerVertices (1GateCircStr (<*x,y*>,'&'))) \/ (InnerVertices (1GateCircStr (<*y,c*>,'&'))))) by CIRCCOMB:49
.= ((((InputVertices (1GateCircStr (<*x,y*>,'&'))) \ {[<*y,c*>,'&']}) \/ ((InputVertices (1GateCircStr (<*y,c*>,'&'))) \ {[<*x,y*>,'&']})) \ (InnerVertices (1GateCircStr (<*c,x*>,'&')))) \/ ((InputVertices (1GateCircStr (<*c,x*>,'&'))) \ ((InnerVertices (1GateCircStr (<*x,y*>,'&'))) \/ (InnerVertices (1GateCircStr (<*y,c*>,'&'))))) by CIRCCOMB:49
.= ((((InputVertices (1GateCircStr (<*x,y*>,'&'))) \ {[<*y,c*>,'&']}) \/ ((InputVertices (1GateCircStr (<*y,c*>,'&'))) \ {[<*x,y*>,'&']})) \ {[<*c,x*>,'&']}) \/ ((InputVertices (1GateCircStr (<*c,x*>,'&'))) \ ((InnerVertices (1GateCircStr (<*x,y*>,'&'))) \/ (InnerVertices (1GateCircStr (<*y,c*>,'&'))))) by CIRCCOMB:49
.= ((((InputVertices (1GateCircStr (<*x,y*>,'&'))) \ {[<*y,c*>,'&']}) \/ ((InputVertices (1GateCircStr (<*y,c*>,'&'))) \ {[<*x,y*>,'&']})) \ {[<*c,x*>,'&']}) \/ ((InputVertices (1GateCircStr (<*c,x*>,'&'))) \ ({[<*x,y*>,'&']} \/ (InnerVertices (1GateCircStr (<*y,c*>,'&'))))) by CIRCCOMB:49
.= ((((InputVertices (1GateCircStr (<*x,y*>,'&'))) \ {[<*y,c*>,'&']}) \/ ((InputVertices (1GateCircStr (<*y,c*>,'&'))) \ {[<*x,y*>,'&']})) \ {[<*c,x*>,'&']}) \/ ((InputVertices (1GateCircStr (<*c,x*>,'&'))) \ ({[<*x,y*>,'&']} \/ {[<*y,c*>,'&']})) by CIRCCOMB:49
.= ((({x,y} \ {[<*y,c*>,'&']}) \/ ((InputVertices (1GateCircStr (<*y,c*>,'&'))) \ {[<*x,y*>,'&']})) \ {[<*c,x*>,'&']}) \/ ((InputVertices (1GateCircStr (<*c,x*>,'&'))) \ ({[<*x,y*>,'&']} \/ {[<*y,c*>,'&']})) by FACIRC_1:40
.= ((({x,y} \ {[<*y,c*>,'&']}) \/ ({y,c} \ {[<*x,y*>,'&']})) \ {[<*c,x*>,'&']}) \/ ((InputVertices (1GateCircStr (<*c,x*>,'&'))) \ ({[<*x,y*>,'&']} \/ {[<*y,c*>,'&']})) by FACIRC_1:40
.= ((({x,y} \ {[<*y,c*>,'&']}) \/ ({y,c} \ {[<*x,y*>,'&']})) \ {[<*c,x*>,'&']}) \/ ({c,x} \ ({[<*x,y*>,'&']} \/ {[<*y,c*>,'&']})) by FACIRC_1:40
.= ((({x,y} \ {[<*y,c*>,'&']}) \/ ({y,c} \ {[<*x,y*>,'&']})) \ {[<*c,x*>,'&']}) \/ ({c,x} \ {[<*x,y*>,'&'],[<*y,c*>,'&']}) by ENUMSET1:41
.= (({x,y} \/ ({y,c} \ {[<*x,y*>,'&']})) \ {[<*c,x*>,'&']}) \/ ({c,x} \ {[<*x,y*>,'&'],[<*y,c*>,'&']}) by A1, A13, Th1
.= (({x,y} \/ {y,c}) \ {[<*c,x*>,'&']}) \/ ({c,x} \ {[<*x,y*>,'&'],[<*y,c*>,'&']}) by A25, ZFMISC_1:65
.= (({x,y} \/ {y,c}) \ {[<*c,x*>,'&']}) \/ {c,x} by A32, A33, ZFMISC_1:72
.= ({x,y,y,c} \ {[<*c,x*>,'&']}) \/ {c,x} by ENUMSET1:45
.= ({y,y,x,c} \ {[<*c,x*>,'&']}) \/ {c,x} by ENUMSET1:110
.= ({y,x,c} \ {[<*c,x*>,'&']}) \/ {c,x} by ENUMSET1:71
.= ({x,y,c} \ {[<*c,x*>,'&']}) \/ {c,x} by ENUMSET1:99
.= {x,y,c} \/ {c,x} by A47, ZFMISC_1:65
.= {x,y,c,c,x} by ENUMSET1:49
.= {x,y,c,c} \/ {x} by ENUMSET1:50
.= {c,c,x,y} \/ {x} by ENUMSET1:118
.= {c,x,y} \/ {x} by ENUMSET1:71
.= {c,x,y,x} by ENUMSET1:46
.= {x,x,y,c} by ENUMSET1:113
.= {x,y,c} by ENUMSET1:71 ;
hence InputVertices (MajorityIStr (x,y,c)) = {x,y,c} ; :: thesis: verum