
theorem Th14:
  for P, Q being pcs-Str holds
  the carrier of pcs-sum(P,Q) = (the carrier of P) \/ the carrier of Q &
  the InternalRel of pcs-sum(P,Q) =
  (the InternalRel of P) \/ the InternalRel of Q &
  the ToleranceRel of pcs-sum(P,Q) =
  (the ToleranceRel of P) \/ the ToleranceRel of Q
proof
  let P, Q be pcs-Str;
  set S = pcsSUM(P,Q);
  set f = <%P,Q%>;
A1: dom Carrier f = {0,1} by PARTFUN1:def 2;
A2: dom pcs-InternalRels f = {0,1} by PARTFUN1:def 2;
A3: dom pcs-ToleranceRels f = {0,1} by PARTFUN1:def 2;
A4: the carrier of S = Union Carrier f
  proof
    thus the carrier of S c= Union Carrier f
    proof
      let x be object;
      assume x in the carrier of S;
      then
A5:   x in the carrier of P or x in the carrier of Q by XBOOLE_0:def 3;
A6:   (Carrier f).z = the carrier of f.z by Def1;
A7:   (Carrier f).j = the carrier of f.j by Def1;
A8:  the carrier of P in rng Carrier f by A1,A6,FUNCT_1:3;
      the carrier of Q in rng Carrier f by A1,A7,FUNCT_1:3;
      hence thesis by A5,A8,TARSKI:def 4;
    end;
    let x be object;
    assume x in Union Carrier f;
    then consider Z being set such that
A9: x in Z and
A10: Z in rng Carrier f by TARSKI:def 4;
    consider i being object such that
A11: i in dom Carrier f and
A12: Carrier f.i = Z by A10,FUNCT_1:def 3;
    i = 0 or i = 1 by A11,TARSKI:def 2;
    then Z = the carrier of f.z or Z = the carrier of f.j by A12,Def1;
    hence thesis by A9,XBOOLE_0:def 3;
  end;
A13: the InternalRel of S = Union pcs-InternalRels f
  proof
    thus the InternalRel of S c= Union pcs-InternalRels f
    proof
      let x be object;
      assume x in the InternalRel of S;
      then
A14:  x in the InternalRel of P or x in the InternalRel of Q by XBOOLE_0:def 3;
A15:  (pcs-InternalRels f).z = the InternalRel of f.z by Def6;
A16:  (pcs-InternalRels f).j = the InternalRel of f.j by Def6;
A17:  the InternalRel of P in rng pcs-InternalRels f by A2,A15,FUNCT_1:3;
      the InternalRel of Q in rng pcs-InternalRels f by A2,A16,FUNCT_1:3;
      hence thesis by A14,A17,TARSKI:def 4;
    end;
    let x be object;
    assume x in Union pcs-InternalRels f;
    then consider Z being set such that
A18: x in Z and
A19: Z in rng pcs-InternalRels f by TARSKI:def 4;
    consider i being object such that
A20: i in dom pcs-InternalRels f and
A21: (pcs-InternalRels f).i = Z by A19,FUNCT_1:def 3;
    i = 0 or i = 1 by A20,TARSKI:def 2;
    then Z = the InternalRel of f.z or Z = the InternalRel of f.j by A21,Def6;
    hence thesis by A18,XBOOLE_0:def 3;
  end;
  the ToleranceRel of S = Union pcs-ToleranceRels f
  proof
    thus the ToleranceRel of S c= Union pcs-ToleranceRels f
    proof
      let x be object;
      assume x in the ToleranceRel of S;
      then
A22:  x in the ToleranceRel of P or x in the ToleranceRel of Q
      by XBOOLE_0:def 3;
A23:  (pcs-ToleranceRels f).z = the ToleranceRel of f.z by Def20;
A24:  (pcs-ToleranceRels f).j = the ToleranceRel of f.j by Def20;
A25:  the ToleranceRel of P in rng pcs-ToleranceRels f by A3,A23,FUNCT_1:3;
      the ToleranceRel of Q in rng pcs-ToleranceRels f by A3,A24,FUNCT_1:3;
      hence thesis by A22,A25,TARSKI:def 4;
    end;
    let x be object;
    assume x in Union pcs-ToleranceRels f;
    then consider Z being set such that
A26: x in Z and
A27: Z in rng pcs-ToleranceRels f by TARSKI:def 4;
    consider i being object such that
A28: i in dom pcs-ToleranceRels f and
A29: (pcs-ToleranceRels f).i = Z by A27,FUNCT_1:def 3;
    i = 0 or i = 1 by A28,TARSKI:def 2;
    then Z = the ToleranceRel of f.z or
    Z = the ToleranceRel of f.j by A29,Def20;
    hence thesis by A26,XBOOLE_0:def 3;
  end;
  hence thesis by A4,A13,Def36;
end;
