
theorem :: ThSTC0IS8:
  for x1,x2,x3,x5,x6,x7 being non pair set
   for x4 being set st
    x4 <> [<*GFA0AdderOutput(x1,x2,x3),GFA0AdderOutput(x5,x6,x7)*>,xor2] &
    x4 <> [<*GFA0AdderOutput(x1,x2,x3),GFA0AdderOutput(x5,x6,x7)*>,and2] &
    not x4 in InnerVertices STC0IIStr(x1,x2,x3,x5,x6,x7)
  holds
    x1 in InputVertices STC0IStr(x1,x2,x3,x4,x5,x6,x7) &
    x2 in InputVertices STC0IStr(x1,x2,x3,x4,x5,x6,x7) &
    x3 in InputVertices STC0IStr(x1,x2,x3,x4,x5,x6,x7) &
    x4 in InputVertices STC0IStr(x1,x2,x3,x4,x5,x6,x7) &
    x5 in InputVertices STC0IStr(x1,x2,x3,x4,x5,x6,x7) &
    x6 in InputVertices STC0IStr(x1,x2,x3,x4,x5,x6,x7) &
    x7 in InputVertices STC0IStr(x1,x2,x3,x4,x5,x6,x7)
  proof
    let x1,x2,x3,x5,x6,x7 be non pair set;
    let x4 be set;
    set S = STC0IStr(x1,x2,x3,x4,x5,x6,x7);
    set S1 = STC0IIStr(x1,x2,x3,x5,x6,x7);
    set A1 = GFA0AdderOutput(x1,x2,x3);
    set A2 = GFA0AdderOutput(x5,x6,x7);
    set A1A20 = [<*A1,A2*>,xor2];
    set A1A2 = [<*A1,A2*>,and2];

    assume x4 <> A1A20 & x4 <> A1A2 & not x4 in InnerVertices S1;
    then InputVertices S = {x1,x2,x3,x4,x5,x6,x7} by ThSTC0IS3;
    hence thesis by ENUMSET1:def 5;
end;
