let x, y, z be set ; :: thesis: ( x <> [<*y,z*>,and2] & y <> [<*z,x*>,and2] & z <> [<*x,y*>,and2] implies InputVertices (GFA0CarryIStr (x,y,z)) = {x,y,z} )
set f1 = and2 ;
set f2 = and2 ;
set f3 = and2 ;
set xy = [<*x,y*>,and2];
set yz = [<*y,z*>,and2];
set zx = [<*z,x*>,and2];
set Cxy = 1GateCircStr (<*x,y*>,and2);
set Cyz = 1GateCircStr (<*y,z*>,and2);
set Czx = 1GateCircStr (<*z,x*>,and2);
assume that
A1: x <> [<*y,z*>,and2] and
A2: ( y <> [<*z,x*>,and2] & z <> [<*x,y*>,and2] ) ; :: thesis: InputVertices (GFA0CarryIStr (x,y,z)) = {x,y,z}
A3: not [<*x,y*>,and2] in {y,z} by A1, A2, Lm1;
A4: not [<*z,x*>,and2] in {x,y,z} by A1, A2, Lm1;
A5: y <> [<*y,z*>,and2] by FACIRC_2:3;
A6: ( not z in {[<*x,y*>,and2],[<*y,z*>,and2]} & not x in {[<*x,y*>,and2],[<*y,z*>,and2]} ) by A1, A2, Lm1;
A7: 1GateCircStr (<*x,y*>,and2) tolerates 1GateCircStr (<*y,z*>,and2) by CIRCCOMB:55;
InputVertices (GFA0CarryIStr (x,y,z)) = ((InputVertices ((1GateCircStr (<*x,y*>,and2)) +* (1GateCircStr (<*y,z*>,and2)))) \ (InnerVertices (1GateCircStr (<*z,x*>,and2)))) \/ ((InputVertices (1GateCircStr (<*z,x*>,and2))) \ (InnerVertices ((1GateCircStr (<*x,y*>,and2)) +* (1GateCircStr (<*y,z*>,and2))))) by CIRCCMB2:6, CIRCCOMB:55
.= ((((InputVertices (1GateCircStr (<*x,y*>,and2))) \ (InnerVertices (1GateCircStr (<*y,z*>,and2)))) \/ ((InputVertices (1GateCircStr (<*y,z*>,and2))) \ (InnerVertices (1GateCircStr (<*x,y*>,and2))))) \ (InnerVertices (1GateCircStr (<*z,x*>,and2)))) \/ ((InputVertices (1GateCircStr (<*z,x*>,and2))) \ (InnerVertices ((1GateCircStr (<*x,y*>,and2)) +* (1GateCircStr (<*y,z*>,and2))))) by CIRCCMB2:6, CIRCCOMB:55
.= ((((InputVertices (1GateCircStr (<*x,y*>,and2))) \ (InnerVertices (1GateCircStr (<*y,z*>,and2)))) \/ ((InputVertices (1GateCircStr (<*y,z*>,and2))) \ (InnerVertices (1GateCircStr (<*x,y*>,and2))))) \ (InnerVertices (1GateCircStr (<*z,x*>,and2)))) \/ ((InputVertices (1GateCircStr (<*z,x*>,and2))) \ ((InnerVertices (1GateCircStr (<*x,y*>,and2))) \/ (InnerVertices (1GateCircStr (<*y,z*>,and2))))) by A7, CIRCCOMB:15
.= ((((InputVertices (1GateCircStr (<*x,y*>,and2))) \ {[<*y,z*>,and2]}) \/ ((InputVertices (1GateCircStr (<*y,z*>,and2))) \ (InnerVertices (1GateCircStr (<*x,y*>,and2))))) \ (InnerVertices (1GateCircStr (<*z,x*>,and2)))) \/ ((InputVertices (1GateCircStr (<*z,x*>,and2))) \ ((InnerVertices (1GateCircStr (<*x,y*>,and2))) \/ (InnerVertices (1GateCircStr (<*y,z*>,and2))))) by CIRCCOMB:49
.= ((((InputVertices (1GateCircStr (<*x,y*>,and2))) \ {[<*y,z*>,and2]}) \/ ((InputVertices (1GateCircStr (<*y,z*>,and2))) \ {[<*x,y*>,and2]})) \ (InnerVertices (1GateCircStr (<*z,x*>,and2)))) \/ ((InputVertices (1GateCircStr (<*z,x*>,and2))) \ ((InnerVertices (1GateCircStr (<*x,y*>,and2))) \/ (InnerVertices (1GateCircStr (<*y,z*>,and2))))) by CIRCCOMB:49
.= ((((InputVertices (1GateCircStr (<*x,y*>,and2))) \ {[<*y,z*>,and2]}) \/ ((InputVertices (1GateCircStr (<*y,z*>,and2))) \ {[<*x,y*>,and2]})) \ {[<*z,x*>,and2]}) \/ ((InputVertices (1GateCircStr (<*z,x*>,and2))) \ ((InnerVertices (1GateCircStr (<*x,y*>,and2))) \/ (InnerVertices (1GateCircStr (<*y,z*>,and2))))) by CIRCCOMB:49
.= ((((InputVertices (1GateCircStr (<*x,y*>,and2))) \ {[<*y,z*>,and2]}) \/ ((InputVertices (1GateCircStr (<*y,z*>,and2))) \ {[<*x,y*>,and2]})) \ {[<*z,x*>,and2]}) \/ ((InputVertices (1GateCircStr (<*z,x*>,and2))) \ ({[<*x,y*>,and2]} \/ (InnerVertices (1GateCircStr (<*y,z*>,and2))))) by CIRCCOMB:49
.= ((((InputVertices (1GateCircStr (<*x,y*>,and2))) \ {[<*y,z*>,and2]}) \/ ((InputVertices (1GateCircStr (<*y,z*>,and2))) \ {[<*x,y*>,and2]})) \ {[<*z,x*>,and2]}) \/ ((InputVertices (1GateCircStr (<*z,x*>,and2))) \ ({[<*x,y*>,and2]} \/ {[<*y,z*>,and2]})) by CIRCCOMB:49
.= ((({x,y} \ {[<*y,z*>,and2]}) \/ ((InputVertices (1GateCircStr (<*y,z*>,and2))) \ {[<*x,y*>,and2]})) \ {[<*z,x*>,and2]}) \/ ((InputVertices (1GateCircStr (<*z,x*>,and2))) \ ({[<*x,y*>,and2]} \/ {[<*y,z*>,and2]})) by FACIRC_1:40
.= ((({x,y} \ {[<*y,z*>,and2]}) \/ ({y,z} \ {[<*x,y*>,and2]})) \ {[<*z,x*>,and2]}) \/ ((InputVertices (1GateCircStr (<*z,x*>,and2))) \ ({[<*x,y*>,and2]} \/ {[<*y,z*>,and2]})) by FACIRC_1:40
.= ((({x,y} \ {[<*y,z*>,and2]}) \/ ({y,z} \ {[<*x,y*>,and2]})) \ {[<*z,x*>,and2]}) \/ ({z,x} \ ({[<*x,y*>,and2]} \/ {[<*y,z*>,and2]})) by FACIRC_1:40
.= ((({x,y} \ {[<*y,z*>,and2]}) \/ ({y,z} \ {[<*x,y*>,and2]})) \ {[<*z,x*>,and2]}) \/ ({z,x} \ {[<*x,y*>,and2],[<*y,z*>,and2]}) by ENUMSET1:41
.= (({x,y} \/ ({y,z} \ {[<*x,y*>,and2]})) \ {[<*z,x*>,and2]}) \/ ({z,x} \ {[<*x,y*>,and2],[<*y,z*>,and2]}) by A1, A5, FACIRC_2:1
.= (({x,y} \/ {y,z}) \ {[<*z,x*>,and2]}) \/ ({z,x} \ {[<*x,y*>,and2],[<*y,z*>,and2]}) by A3, ZFMISC_1:65
.= (({x,y} \/ {y,z}) \ {[<*z,x*>,and2]}) \/ {z,x} by A6, ZFMISC_1:72
.= ({x,y,y,z} \ {[<*z,x*>,and2]}) \/ {z,x} by ENUMSET1:45
.= ({y,y,x,z} \ {[<*z,x*>,and2]}) \/ {z,x} by ENUMSET1:110
.= ({y,x,z} \ {[<*z,x*>,and2]}) \/ {z,x} by ENUMSET1:71
.= ({x,y,z} \ {[<*z,x*>,and2]}) \/ {z,x} by ENUMSET1:99
.= {x,y,z} \/ {z,x} by A4, ZFMISC_1:65
.= {x,y,z,z,x} by ENUMSET1:49
.= {x,y,z,z} \/ {x} by ENUMSET1:50
.= {z,z,x,y} \/ {x} by ENUMSET1:118
.= {z,x,y} \/ {x} by ENUMSET1:71
.= {z,x,y,x} by ENUMSET1:46
.= {x,x,y,z} by ENUMSET1:113
.= {x,y,z} by ENUMSET1:71 ;
hence InputVertices (GFA0CarryIStr (x,y,z)) = {x,y,z} ; :: thesis: verum