let x, y, z be set ; :: thesis: ( x <> [<*y,z*>,and2c] & y <> [<*z,x*>,nor2] & z <> [<*x,y*>,and2a] implies InputVertices (GFA2CarryStr (x,y,z)) = {x,y,z} )
set f1 = and2a ;
set f2 = and2c ;
set f3 = nor2 ;
set f4 = nor3 ;
set xy = [<*x,y*>,and2a];
set yz = [<*y,z*>,and2c];
set zx = [<*z,x*>,nor2];
set xyz = [<*[<*x,y*>,and2a],[<*y,z*>,and2c],[<*z,x*>,nor2]*>,nor3];
set S = 1GateCircStr (<*[<*x,y*>,and2a],[<*y,z*>,and2c],[<*z,x*>,nor2]*>,nor3);
set MI = GFA2CarryIStr (x,y,z);
A1: InputVertices (1GateCircStr (<*[<*x,y*>,and2a],[<*y,z*>,and2c],[<*z,x*>,nor2]*>,nor3)) = rng <*[<*x,y*>,and2a],[<*y,z*>,and2c],[<*z,x*>,nor2]*> by CIRCCOMB:42
.= {[<*x,y*>,and2a],[<*y,z*>,and2c],[<*z,x*>,nor2]} by FINSEQ_2:128 ;
assume A2: ( x <> [<*y,z*>,and2c] & y <> [<*z,x*>,nor2] & z <> [<*x,y*>,and2a] ) ; :: thesis: InputVertices (GFA2CarryStr (x,y,z)) = {x,y,z}
A3: ( InnerVertices (1GateCircStr (<*[<*x,y*>,and2a],[<*y,z*>,and2c],[<*z,x*>,nor2]*>,nor3)) = {[<*[<*x,y*>,and2a],[<*y,z*>,and2c],[<*z,x*>,nor2]*>,nor3]} & {x,y,z} \ {[<*[<*x,y*>,and2a],[<*y,z*>,and2c],[<*z,x*>,nor2]*>,nor3]} = {x,y,z} ) by Lm2, CIRCCOMB:42;
A4: {[<*x,y*>,and2a],[<*y,z*>,and2c],[<*z,x*>,nor2]} \ {[<*x,y*>,and2a],[<*y,z*>,and2c],[<*z,x*>,nor2]} = {} by XBOOLE_1:37;
thus InputVertices (GFA2CarryStr (x,y,z)) = ((InputVertices (GFA2CarryIStr (x,y,z))) \ (InnerVertices (1GateCircStr (<*[<*x,y*>,and2a],[<*y,z*>,and2c],[<*z,x*>,nor2]*>,nor3)))) \/ ((InputVertices (1GateCircStr (<*[<*x,y*>,and2a],[<*y,z*>,and2c],[<*z,x*>,nor2]*>,nor3))) \ (InnerVertices (GFA2CarryIStr (x,y,z)))) by CIRCCMB2:5, CIRCCOMB:47
.= {x,y,z} \/ ({[<*x,y*>,and2a],[<*y,z*>,and2c],[<*z,x*>,nor2]} \ (InnerVertices (GFA2CarryIStr (x,y,z)))) by A1, A2, A3, Th76
.= {x,y,z} \/ {} by A4, Th73
.= {x,y,z} ; :: thesis: verum