let x, y, z be set ; :: thesis: ( x <> [<*y,z*>,and2a] & y <> [<*z,x*>,and2] & z <> [<*x,y*>,and2c] implies InputVertices (GFA1CarryStr (x,y,z)) = {x,y,z} )
set f1 = and2c ;
set f2 = and2a ;
set f3 = and2 ;
set f4 = or3 ;
set xy = [<*x,y*>,and2c];
set yz = [<*y,z*>,and2a];
set zx = [<*z,x*>,and2];
set xyz = [<*[<*x,y*>,and2c],[<*y,z*>,and2a],[<*z,x*>,and2]*>,or3];
set S = 1GateCircStr (<*[<*x,y*>,and2c],[<*y,z*>,and2a],[<*z,x*>,and2]*>,or3);
set MI = GFA1CarryIStr (x,y,z);
A1: InputVertices (1GateCircStr (<*[<*x,y*>,and2c],[<*y,z*>,and2a],[<*z,x*>,and2]*>,or3)) = rng <*[<*x,y*>,and2c],[<*y,z*>,and2a],[<*z,x*>,and2]*> by CIRCCOMB:42
.= {[<*x,y*>,and2c],[<*y,z*>,and2a],[<*z,x*>,and2]} by FINSEQ_2:128 ;
assume A2: ( x <> [<*y,z*>,and2a] & y <> [<*z,x*>,and2] & z <> [<*x,y*>,and2c] ) ; :: thesis: InputVertices (GFA1CarryStr (x,y,z)) = {x,y,z}
A3: ( InnerVertices (1GateCircStr (<*[<*x,y*>,and2c],[<*y,z*>,and2a],[<*z,x*>,and2]*>,or3)) = {[<*[<*x,y*>,and2c],[<*y,z*>,and2a],[<*z,x*>,and2]*>,or3]} & {x,y,z} \ {[<*[<*x,y*>,and2c],[<*y,z*>,and2a],[<*z,x*>,and2]*>,or3]} = {x,y,z} ) by Lm2, CIRCCOMB:42;
A4: {[<*x,y*>,and2c],[<*y,z*>,and2a],[<*z,x*>,and2]} \ {[<*x,y*>,and2c],[<*y,z*>,and2a],[<*z,x*>,and2]} = {} by XBOOLE_1:37;
thus InputVertices (GFA1CarryStr (x,y,z)) = ((InputVertices (GFA1CarryIStr (x,y,z))) \ (InnerVertices (1GateCircStr (<*[<*x,y*>,and2c],[<*y,z*>,and2a],[<*z,x*>,and2]*>,or3)))) \/ ((InputVertices (1GateCircStr (<*[<*x,y*>,and2c],[<*y,z*>,and2a],[<*z,x*>,and2]*>,or3))) \ (InnerVertices (GFA1CarryIStr (x,y,z)))) by CIRCCMB2:5, CIRCCOMB:47
.= {x,y,z} \/ ({[<*x,y*>,and2c],[<*y,z*>,and2a],[<*z,x*>,and2]} \ (InnerVertices (GFA1CarryIStr (x,y,z)))) by A1, A2, A3, Th44
.= {x,y,z} \/ {} by A4, Th41
.= {x,y,z} ; :: thesis: verum