let x, y, c be set ; :: thesis:
assume A1: ( x <> [<*y,c*>,and2 ] & y <> [<*x,c*>,and2a ] & c <> [<*x,y*>,and2a ] ) ; :: thesis:
A2: 1GateCircStr <*x,y*>,and2a tolerates 1GateCircStr <*y,c*>,and2 by CIRCCOMB:55;
A3: ( y in {1,y} & y in {2,y} ) by TARSKI:def 2;
A4: ( {1,y} in [1,y] & {2,y} in [2,y] ) by TARSKI:def 2;
<*y,c*> = <*y*> ^ <*c*> by FINSEQ_1:def 9;
then A5: <*y*> c= <*y,c*> by FINSEQ_6:12;
<*y*> = {[1,y]} by FINSEQ_1:def 5;
then A6: [1,y] in <*y*> by TARSKI:def 1;
A7: ( <*y,c*> in {<*y,c*>} & {<*y,c*>} in [<*y,c*>,and2 ] ) by TARSKI:def 1, TARSKI:def 2;
then A8: y <> [<*y,c*>,and2 ] by A3, A4, A5, A6, ORDINAL1:5;
A9: c in {2,c} by TARSKI:def 2;
A10: {2,c} in [2,c] by TARSKI:def 2;
( 2 in dom <*y,c*> & <*y,c*> . 2 = c )

dom <*y*> = Seg 1 by FINSEQ_1:def 8;
then A11: len <*y*> = 1 by FINSEQ_1:def 3;
dom <*c*> = Seg 1 by FINSEQ_1:def 8;
then A12: len <*c*> = 1 by FINSEQ_1:def 3;
dom <*y,c*> = dom (<*y*> ^ <*c*>) by FINSEQ_1:def 9
.= Seg 2 by A11, A12, FINSEQ_1:def 7 ;
hence ( 2 in dom <*y,c*> & <*y,c*> . 2 = c ) by FINSEQ_1:3, FINSEQ_1:61; :: thesis:

end;

then [2,c] in <*y,c*> by FUNCT_1:8;
then A13: c <> [<*y,c*>,and2 ] by A7, A9, A10, ORDINAL1:5;
( 2 in dom <*x,y*> & <*x,y*> . 2 = y )

dom <*x*> = Seg 1 by FINSEQ_1:def 8;
then A14: len <*x*> = 1 by FINSEQ_1:def 3;
dom <*y*> = Seg 1 by FINSEQ_1:def 8;
then A15: len <*y*> = 1 by FINSEQ_1:def 3;
dom <*x,y*> = dom (<*x*> ^ <*y*>) by FINSEQ_1:def 9
.= Seg 2 by A14, A15, FINSEQ_1:def 7 ;
hence ( 2 in dom <*x,y*> & <*x,y*> . 2 = y ) by FINSEQ_1:3, FINSEQ_1:61; :: thesis:

end;

then A16: [2,y] in <*x,y*> by FUNCT_1:8;
( <*x,y*> in {<*x,y*>} & {<*x,y*>} in [<*x,y*>,and2a ] ) by TARSKI:def 1, TARSKI:def 2;
then y <> [<*x,y*>,and2a ] by A3, A4, A16, ORDINAL1:5;
then A17: not [<*x,y*>,and2a ] in {y,c} by A1, TARSKI:def 2;
A18: x in {1,x} by TARSKI:def 2;
A19: {1,x} in [1,x] by TARSKI:def 2;
<*x,y*> = <*x*> ^ <*y*> by FINSEQ_1:def 9;
then A20: <*x*> c= <*x,y*> by FINSEQ_6:12;
<*x*> = {[1,x]} by FINSEQ_1:def 5;
then A21: [1,x] in <*x*> by TARSKI:def 1;
( <*x,y*> in {<*x,y*>} & {<*x,y*>} in [<*x,y*>,and2a ] ) by TARSKI:def 1, TARSKI:def 2;
then x <> [<*x,y*>,and2a ] by A18, A19, A20, A21, ORDINAL1:5;
then A22: ( not c in {[<*x,y*>,and2a ],[<*y,c*>,and2 ]} & not x in {[<*x,y*>,and2a ],[<*y,c*>,and2 ]} ) by A1, A13, TARSKI:def 2;
A23: c in {2,c} by TARSKI:def 2;
A24: {2,c} in [2,c] by TARSKI:def 2;
( 2 in dom <*x,c*> & <*x,c*> . 2 = c )

dom <*c*> = Seg 1 by FINSEQ_1:def 8;
then A25: len <*c*> = 1 by FINSEQ_1:def 3;
dom <*x*> = Seg 1 by FINSEQ_1:def 8;
then A26: len <*x*> = 1 by FINSEQ_1:def 3;
dom <*x,c*> = dom (<*x*> ^ <*c*>) by FINSEQ_1:def 9
.= Seg 2 by A25, A26, FINSEQ_1:def 7 ;
hence ( 2 in dom <*x,c*> & <*x,c*> . 2 = c ) by FINSEQ_1:3, FINSEQ_1:61; :: thesis:

end;