let x1, x2, x3, x5, x6, x7 be set ; :: thesis: ( [<*x1,x2*>,xor2] in InnerVertices (STC0IIStr (x1,x2,x3,x5,x6,x7)) & GFA0AdderOutput (x1,x2,x3) in InnerVertices (STC0IIStr (x1,x2,x3,x5,x6,x7)) & [<*x1,x2*>,and2] in InnerVertices (STC0IIStr (x1,x2,x3,x5,x6,x7)) & [<*x2,x3*>,and2] in InnerVertices (STC0IIStr (x1,x2,x3,x5,x6,x7)) & [<*x3,x1*>,and2] in InnerVertices (STC0IIStr (x1,x2,x3,x5,x6,x7)) & GFA0CarryOutput (x1,x2,x3) in InnerVertices (STC0IIStr (x1,x2,x3,x5,x6,x7)) & [<*x5,x6*>,xor2] in InnerVertices (STC0IIStr (x1,x2,x3,x5,x6,x7)) & GFA0AdderOutput (x5,x6,x7) in InnerVertices (STC0IIStr (x1,x2,x3,x5,x6,x7)) & [<*x5,x6*>,and2] in InnerVertices (STC0IIStr (x1,x2,x3,x5,x6,x7)) & [<*x6,x7*>,and2] in InnerVertices (STC0IIStr (x1,x2,x3,x5,x6,x7)) & [<*x7,x5*>,and2] in InnerVertices (STC0IIStr (x1,x2,x3,x5,x6,x7)) & GFA0CarryOutput (x5,x6,x7) in InnerVertices (STC0IIStr (x1,x2,x3,x5,x6,x7)) )
set S = STC0IIStr (x1,x2,x3,x5,x6,x7);
set S1 = BitGFA0Str (x1,x2,x3);
set A1 = GFA0AdderOutput (x1,x2,x3);
set C1 = GFA0CarryOutput (x1,x2,x3);
set S2 = BitGFA0Str (x5,x6,x7);
set A2 = GFA0AdderOutput (x5,x6,x7);
set C2 = GFA0CarryOutput (x5,x6,x7);
set x1x20 = [<*x1,x2*>,xor2];
set x1x2 = [<*x1,x2*>,and2];
set x2x3 = [<*x2,x3*>,and2];
set x3x1 = [<*x3,x1*>,and2];
set x5x60 = [<*x5,x6*>,xor2];
set x5x6 = [<*x5,x6*>,and2];
set x6x7 = [<*x6,x7*>,and2];
set x7x5 = [<*x7,x5*>,and2];
set p1 = {[<*x1,x2*>,xor2],(GFA0AdderOutput (x1,x2,x3)),[<*x1,x2*>,and2],[<*x2,x3*>,and2],[<*x3,x1*>,and2],(GFA0CarryOutput (x1,x2,x3))};
set p2 = {[<*x5,x6*>,xor2],(GFA0AdderOutput (x5,x6,x7)),[<*x5,x6*>,and2],[<*x6,x7*>,and2],[<*x7,x5*>,and2],(GFA0CarryOutput (x5,x6,x7))};
A1: ( [<*x1,x2*>,xor2] in {[<*x1,x2*>,xor2],(GFA0AdderOutput (x1,x2,x3)),[<*x1,x2*>,and2],[<*x2,x3*>,and2],[<*x3,x1*>,and2],(GFA0CarryOutput (x1,x2,x3))} & GFA0AdderOutput (x1,x2,x3) in {[<*x1,x2*>,xor2],(GFA0AdderOutput (x1,x2,x3)),[<*x1,x2*>,and2],[<*x2,x3*>,and2],[<*x3,x1*>,and2],(GFA0CarryOutput (x1,x2,x3))} & [<*x1,x2*>,and2] in {[<*x1,x2*>,xor2],(GFA0AdderOutput (x1,x2,x3)),[<*x1,x2*>,and2],[<*x2,x3*>,and2],[<*x3,x1*>,and2],(GFA0CarryOutput (x1,x2,x3))} & [<*x2,x3*>,and2] in {[<*x1,x2*>,xor2],(GFA0AdderOutput (x1,x2,x3)),[<*x1,x2*>,and2],[<*x2,x3*>,and2],[<*x3,x1*>,and2],(GFA0CarryOutput (x1,x2,x3))} & [<*x3,x1*>,and2] in {[<*x1,x2*>,xor2],(GFA0AdderOutput (x1,x2,x3)),[<*x1,x2*>,and2],[<*x2,x3*>,and2],[<*x3,x1*>,and2],(GFA0CarryOutput (x1,x2,x3))} & GFA0CarryOutput (x1,x2,x3) in {[<*x1,x2*>,xor2],(GFA0AdderOutput (x1,x2,x3)),[<*x1,x2*>,and2],[<*x2,x3*>,and2],[<*x3,x1*>,and2],(GFA0CarryOutput (x1,x2,x3))} ) by ENUMSET1:def 4;
A2: ( [<*x5,x6*>,xor2] in {[<*x5,x6*>,xor2],(GFA0AdderOutput (x5,x6,x7)),[<*x5,x6*>,and2],[<*x6,x7*>,and2],[<*x7,x5*>,and2],(GFA0CarryOutput (x5,x6,x7))} & GFA0AdderOutput (x5,x6,x7) in {[<*x5,x6*>,xor2],(GFA0AdderOutput (x5,x6,x7)),[<*x5,x6*>,and2],[<*x6,x7*>,and2],[<*x7,x5*>,and2],(GFA0CarryOutput (x5,x6,x7))} & [<*x5,x6*>,and2] in {[<*x5,x6*>,xor2],(GFA0AdderOutput (x5,x6,x7)),[<*x5,x6*>,and2],[<*x6,x7*>,and2],[<*x7,x5*>,and2],(GFA0CarryOutput (x5,x6,x7))} & [<*x6,x7*>,and2] in {[<*x5,x6*>,xor2],(GFA0AdderOutput (x5,x6,x7)),[<*x5,x6*>,and2],[<*x6,x7*>,and2],[<*x7,x5*>,and2],(GFA0CarryOutput (x5,x6,x7))} & [<*x7,x5*>,and2] in {[<*x5,x6*>,xor2],(GFA0AdderOutput (x5,x6,x7)),[<*x5,x6*>,and2],[<*x6,x7*>,and2],[<*x7,x5*>,and2],(GFA0CarryOutput (x5,x6,x7))} & GFA0CarryOutput (x5,x6,x7) in {[<*x5,x6*>,xor2],(GFA0AdderOutput (x5,x6,x7)),[<*x5,x6*>,and2],[<*x6,x7*>,and2],[<*x7,x5*>,and2],(GFA0CarryOutput (x5,x6,x7))} ) by ENUMSET1:def 4;
InnerVertices (STC0IIStr (x1,x2,x3,x5,x6,x7)) = (({[<*x1,x2*>,xor2],(GFA0AdderOutput (x1,x2,x3))} \/ {[<*x1,x2*>,and2],[<*x2,x3*>,and2],[<*x3,x1*>,and2],(GFA0CarryOutput (x1,x2,x3))}) \/ {[<*x5,x6*>,xor2],(GFA0AdderOutput (x5,x6,x7))}) \/ {[<*x5,x6*>,and2],[<*x6,x7*>,and2],[<*x7,x5*>,and2],(GFA0CarryOutput (x5,x6,x7))} by ThSTC0IIS1
.= ({[<*x1,x2*>,xor2],(GFA0AdderOutput (x1,x2,x3)),[<*x1,x2*>,and2],[<*x2,x3*>,and2],[<*x3,x1*>,and2],(GFA0CarryOutput (x1,x2,x3))} \/ {[<*x5,x6*>,xor2],(GFA0AdderOutput (x5,x6,x7))}) \/ {[<*x5,x6*>,and2],[<*x6,x7*>,and2],[<*x7,x5*>,and2],(GFA0CarryOutput (x5,x6,x7))} by ENUMSET1:12
.= {[<*x1,x2*>,xor2],(GFA0AdderOutput (x1,x2,x3)),[<*x1,x2*>,and2],[<*x2,x3*>,and2],[<*x3,x1*>,and2],(GFA0CarryOutput (x1,x2,x3))} \/ ({[<*x5,x6*>,xor2],(GFA0AdderOutput (x5,x6,x7))} \/ {[<*x5,x6*>,and2],[<*x6,x7*>,and2],[<*x7,x5*>,and2],(GFA0CarryOutput (x5,x6,x7))}) by XBOOLE_1:4
.= {[<*x1,x2*>,xor2],(GFA0AdderOutput (x1,x2,x3)),[<*x1,x2*>,and2],[<*x2,x3*>,and2],[<*x3,x1*>,and2],(GFA0CarryOutput (x1,x2,x3))} \/ {[<*x5,x6*>,xor2],(GFA0AdderOutput (x5,x6,x7)),[<*x5,x6*>,and2],[<*x6,x7*>,and2],[<*x7,x5*>,and2],(GFA0CarryOutput (x5,x6,x7))} by ENUMSET1:12 ;
hence ( [<*x1,x2*>,xor2] in InnerVertices (STC0IIStr (x1,x2,x3,x5,x6,x7)) & GFA0AdderOutput (x1,x2,x3) in InnerVertices (STC0IIStr (x1,x2,x3,x5,x6,x7)) & [<*x1,x2*>,and2] in InnerVertices (STC0IIStr (x1,x2,x3,x5,x6,x7)) & [<*x2,x3*>,and2] in InnerVertices (STC0IIStr (x1,x2,x3,x5,x6,x7)) & [<*x3,x1*>,and2] in InnerVertices (STC0IIStr (x1,x2,x3,x5,x6,x7)) & GFA0CarryOutput (x1,x2,x3) in InnerVertices (STC0IIStr (x1,x2,x3,x5,x6,x7)) & [<*x5,x6*>,xor2] in InnerVertices (STC0IIStr (x1,x2,x3,x5,x6,x7)) & GFA0AdderOutput (x5,x6,x7) in InnerVertices (STC0IIStr (x1,x2,x3,x5,x6,x7)) & [<*x5,x6*>,and2] in InnerVertices (STC0IIStr (x1,x2,x3,x5,x6,x7)) & [<*x6,x7*>,and2] in InnerVertices (STC0IIStr (x1,x2,x3,x5,x6,x7)) & [<*x7,x5*>,and2] in InnerVertices (STC0IIStr (x1,x2,x3,x5,x6,x7)) & GFA0CarryOutput (x5,x6,x7) in InnerVertices (STC0IIStr (x1,x2,x3,x5,x6,x7)) ) by A1, A2, XBOOLE_0:def 3; :: thesis: verum