let x, y, z be set ; ( 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] )
; 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_2:2;
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:47;
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 CIRCCMB2:5, CIRCCOMB:47
.=
((((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 CIRCCMB2:5, CIRCCOMB:47
.=
((((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:11
.=
((((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:42
.=
((((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:42
.=
((((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:42
.=
((((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:42
.=
((((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:42
.=
((({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 ENUMSET1:1
.=
(({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_1:57
.=
(({x,y} \/ {y,z}) \ {[<*z,x*>,and2]}) \/ {z,x}
by A6, ZFMISC_1:63
.=
({x,y,y,z} \ {[<*z,x*>,and2]}) \/ {z,x}
by ENUMSET1:5
.=
({y,y,x,z} \ {[<*z,x*>,and2]}) \/ {z,x}
by ENUMSET1:67
.=
({y,x,z} \ {[<*z,x*>,and2]}) \/ {z,x}
by ENUMSET1:31
.=
({x,y,z} \ {[<*z,x*>,and2]}) \/ {z,x}
by ENUMSET1:58
.=
{x,y,z} \/ {z,x}
by A4, ZFMISC_1:57
.=
{x,y,z,z,x}
by ENUMSET1:9
.=
{x,y,z,z} \/ {x}
by ENUMSET1:10
.=
{z,z,x,y} \/ {x}
by ENUMSET1:73
.=
{z,x,y} \/ {x}
by ENUMSET1:31
.=
{z,x,y,x}
by ENUMSET1:6
.=
{x,x,y,z}
by ENUMSET1:70
.=
{x,y,z}
by ENUMSET1:31
;
hence
InputVertices (GFA0CarryIStr (x,y,z)) = {x,y,z}
; verum