reserve A,B,C for Ordinal,
  K,L,M,N for Cardinal,
  x,y,y1,y2,z,u for object,X,Y,Z,Z1,Z2 for set,
  n for Nat,
  f,f1,g,h for Function,
  Q,R for Relation;
reserve ff for Cardinal-Function;
reserve F,G for Cardinal-Function;
reserve A,B for set;

theorem Th62:
  A c= sproduct f &
  (for h1,h2 being Function st h1 in A & h2 in A holds h1 tolerates h2)
  implies union A in sproduct f
proof
  assume that
A1: A c= sproduct f and
A2: for h1,h2 being Function st h1 in A & h2 in A holds h1 tolerates h2;
  reconsider g = union A as Function by A1,A2,PARTFUN1:78;
A3: dom g c= dom f
  proof
    let x be object;
    assume x in dom g;
    then consider y being object such that
A4: [x,y] in g by XTUPLE_0:def 12;
    consider h being set such that
A5: [x,y] in h and
A6: h in A by A4,TARSKI:def 4;
    reconsider h as Function by A1,A6;
A7: x in dom h by A5,XTUPLE_0:def 12;
    dom h c= dom f by A1,A6,Th49;
    hence thesis by A7;
  end;
  now
    let x be object;
    assume x in dom g;
    then consider y being object such that
A8: [x,y] in g by XTUPLE_0:def 12;
    consider h being set such that
A9: [x,y] in h and
A10: h in A by A8,TARSKI:def 4;
    reconsider h as Function by A1,A10;
A11: x in dom h by A9,XTUPLE_0:def 12;
    h.x = y by A9,FUNCT_1:1
      .= g.x by A8,FUNCT_1:1;
    hence g.x in f.x by A1,A10,A11,Th49;
  end;
  hence thesis by A3,Def9;
end;
