
theorem Th4:
  for S1,S2 being non empty ManySortedSign st S1 tolerates S2 &
  InnerVertices S2 misses InputVertices S1 holds InputVertices S1 c=
  InputVertices (S1+*S2) & InputVertices (S1+*S2) = (InputVertices S1) \/ (
  InputVertices S2 \ InnerVertices S1)
proof
  let S1,S2 be non empty ManySortedSign;
  set S = S1+*S2;
  set R = the ResultSort of S;
  set R1 = the ResultSort of S1;
  set R2 = the ResultSort of S2;
  assume that
A1: S1 tolerates S2 and
A2: InnerVertices S2 misses InputVertices S1;
A3: InnerVertices S = rng R;
  InnerVertices S1 = rng R1 & InnerVertices S2 = rng R2;
  then
A4: rng R = (rng R1) \/ rng R2 by A1,A3,CIRCCOMB:11;
A5: the carrier of S = (the carrier of S1) \/ the carrier of S2 by
CIRCCOMB:def 2;
  hereby
    let x be object;
    assume
A6: x in InputVertices S1;
    then ( not x in rng R1)& not x in rng R2 by A2,XBOOLE_0:3,def 5;
    then
A7: not x in rng R by A4,XBOOLE_0:def 3;
    x in the carrier of S by A5,A6,XBOOLE_0:def 3;
    hence x in InputVertices S by A7,XBOOLE_0:def 5;
  end;
A8: InputVertices S c= (InputVertices S1) \/ InputVertices S2 by A1,CIRCCOMB:11
;
  hereby
    let x be object;
    assume
A9: x in InputVertices (S1+*S2);
    then not x in rng R by XBOOLE_0:def 5;
    then x in InputVertices S1 or x in InputVertices S2 & not x in
    InnerVertices S1 by A4,A8,A9,XBOOLE_0:def 3;
    then x in InputVertices S1 or x in InputVertices S2 \ InnerVertices S1 by
XBOOLE_0:def 5;
    hence x in (InputVertices S1) \/ (InputVertices S2 \ InnerVertices S1) by
XBOOLE_0:def 3;
  end;
  let x be object;
  assume x in (InputVertices S1) \/ (InputVertices S2 \ InnerVertices S1);
  then
A10: x in InputVertices S1 or x in InputVertices S2 \ rng R1 by XBOOLE_0:def 3;
  then
A11: x in the carrier of S by A5,XBOOLE_0:def 3;
  x in InputVertices S1 & not x in rng R2 or x in InputVertices S2 & not
  x in rng R1 by A2,A10,XBOOLE_0:3,def 5;
  then
A12: not x in rng R2 by XBOOLE_0:def 5;
  not x in rng R1 by A10,XBOOLE_0:def 5;
  then not x in rng R by A4,A12,XBOOLE_0:def 3;
  hence thesis by A11,XBOOLE_0:def 5;
end;
