
theorem ThSTC0IIS7:
  for x1,x2,x3,x5,x6,x7 being set holds
   [<*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)
  proof
    let x1,x2,x3,x5,x6,x7 be set;
    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 = {x1x20,A1,x1x2,x2x3,x3x1,C1};
    set p2 = {x5x60,A2,x5x6,x6x7,x7x5,C2};

A1: x1x20 in p1 & A1 in p1 &
    x1x2 in p1 & x2x3 in p1 & x3x1 in p1 & C1 in p1 by ENUMSET1:def 4;
A2: x5x60 in p2 & A2 in p2 &
    x5x6 in p2 & x6x7 in p2 & x7x5 in p2 & C2 in p2 by ENUMSET1:def 4;

  InnerVertices S
    = {x1x20,A1} \/ {x1x2,x2x3,x3x1,C1} \/ {x5x60,A2} \/ {x5x6,x6x7,x7x5,C2}
      by ThSTC0IIS1
   .= p1 \/ {x5x60,A2} \/ {x5x6,x6x7,x7x5,C2} by ENUMSET1:12
   .= p1 \/ ({x5x60,A2} \/ {x5x6,x6x7,x7x5,C2}) by XBOOLE_1:4
   .= p1 \/ p2 by ENUMSET1:12;
  hence thesis by A1,A2,XBOOLE_0:def 3;
end;
