
theorem ThSTC0IS3:
  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
    InputVertices STC0IStr(x1,x2,x3,x4,x5,x6,x7) = {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 C1 = GFA0CarryOutput(x1,x2,x3);
    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 S2 = BitGFA0Str(A1,A2,x4);
    set A1A20 = [<*A1,A2*>,xor2];
    set A1A2 = [<*A1,A2*>,and2];
    set A2x4 = [<*A2,x4*>,and2];
    set x4A1 = [<*x4,A1*>,and2];

    assume that
A0: x4 <> A1A20 and
A1: x4 <> A1A2 and
A2: not x4 in InnerVertices S1;
A5: InnerVertices S1
    = {x1x20,A1} \/ {x1x2,x2x3,x3x1,C1} \/ {x5x60,A2} \/ {x5x6,x6x7,x7x5,C2}
      by ThSTC0IIS1
   .= ({A1,x1x20} \/ {A2,x5x60}) \/ {x1x2,x2x3,x3x1,C1} \/ {x5x6,x6x7,x7x5,C2}
      by XBOOLE_1:4
   .= ({A1,x1x20,A2,x5x60}) \/ {x1x2,x2x3,x3x1,C1} \/ {x5x6,x6x7,x7x5,C2}
      by ENUMSET1:5
   .= ({A1,A2,x1x20,x5x60}) \/ {x1x2,x2x3,x3x1,C1} \/ {x5x6,x6x7,x7x5,C2}
      by ENUMSET1:62
   .= ({A1,A2} \/ {x1x20,x5x60}) \/ {x1x2,x2x3,x3x1,C1} \/ {x5x6,x6x7,x7x5,C2}
      by ENUMSET1:5
   .= {A1,A2} \/ ({x1x20,x5x60} \/ {x1x2,x2x3,x3x1,C1}) \/ {x5x6,x6x7,x7x5,C2}
      by XBOOLE_1:4
   .= {A1,A2} \/
      ({x1x20} \/ {x5x60} \/ {x1x2,x2x3,x3x1,C1}) \/ {x5x6,x6x7,x7x5,C2}
      by ENUMSET1:1
   .= {A1,A2} \/
      ({x1x20} \/ ({x5x60} \/ {x1x2,x2x3,x3x1,C1})) \/ {x5x6,x6x7,x7x5,C2}
      by XBOOLE_1:4
   .= {A1,A2} \/ ({x1x20} \/ {x5x60,x1x2,x2x3,x3x1,C1}) \/ {x5x6,x6x7,x7x5,C2}
      by ENUMSET1:7
   .= {A1,A2} \/
      (({x1x20} \/ {x5x60,x1x2,x2x3,x3x1,C1}) \/ {x5x6,x6x7,x7x5,C2})
      by XBOOLE_1:4
   .= {A1,A2} \/
      ({x1x20} \/ ({x5x60,x1x2,x2x3,x3x1,C1} \/ {x5x6,x6x7,x7x5,C2}))
      by XBOOLE_1:4
   .= {A1,A2} \/ ({x1x20} \/ {x5x60,x1x2,x2x3,x3x1,C1,x5x6,x6x7,x7x5,C2})
      by ENUMSET1:81
   .= {A1,A2} \/ {x5x60,x1x2,x2x3,x3x1,C1,x5x6,x6x7,x7x5,C2,x1x20}
      by ENUMSET1:85;
A6: {A1,A2,x4} \ InnerVertices S1
    = ({A1,A2} \/ {x4}) \ InnerVertices S1 by ENUMSET1:3
   .= ({A1,A2} \ InnerVertices S1) \/ ({x4} \ InnerVertices S1) by XBOOLE_1:42
   .= ({A1,A2} \ InnerVertices S1) \/ ({x4}) by A2,ZFMISC_1:59
   .= ({}) \/ ({x4}) by A5,XBOOLE_1:46
   .= {x4};
A7: A1 <> A2x4 & A2 <> x4A1 by LmSTC0IS1;
    InnerVertices S2 misses InputVertices S1 & S1 tolerates S2
    by LmSTC0IS2b,CIRCCOMB:47;
    hence
    InputVertices (S)
      = (InputVertices S1) \/ (InputVertices S2 \ InnerVertices S1)
        by FACIRC_1:4
     .= ({x1,x2,x3,x5,x6,x7}) \/ (InputVertices S2 \ InnerVertices S1)
        by ThSTC0IIS4
     .= ({x1,x2,x3,x5,x6,x7}) \/ ({x4}) by A0,A1,A7,GFACIRC1:33,A6
     .= ({x1,x2,x3} \/ {x5,x6,x7}) \/ ({x4}) by ENUMSET1:13
     .= ({x1,x2,x3}) \/ ({x5,x6,x7} \/ ({x4})) by XBOOLE_1:4
     .= ({x1,x2,x3}) \/ ({x4,x5,x6,x7}) by ENUMSET1:4
     .= {x1,x2,x3,x4,x5,x6,x7} by ENUMSET1:18;
  end;
