reserve r for Real,
  X for set,
  f, g, h for real-valued Function;

theorem Th14:
  for X, Y being non empty TopSpace, X1, X2 being non empty
  SubSpace of X for f1 being Function of X1,Y, f2 being Function of X2,Y st X1
misses X2 or f1|(X1 meet X2) = f2|(X1 meet X2) holds (for A being Subset of X1
holds (f1 union f2).:A = f1.:A) & for A being Subset of X2 holds (f1 union f2)
  .:A = f2.:A
proof
  let X, Y be non empty TopSpace, X1, X2 be non empty SubSpace of X;
  let f1 be Function of X1,Y, f2 be Function of X2,Y;
  assume
A1: X1 misses X2 or f1|(X1 meet X2) = f2|(X1 meet X2);
  set F = f1 union f2;
A2: the carrier of X1 union X2 = (the carrier of X1) \/ the carrier of X2 by
TSEP_1:def 2;
  hereby
    let A be Subset of X1;
    thus (f1 union f2).:A = f1.:A
    proof
      hereby
        let y be object;
        assume y in (f1 union f2).:A;
        then consider x being Element of X1 union X2 such that
A3:     x in A and
A4:     y = F.x by FUNCT_2:65;
        x is Point of X by PRE_TOPC:25;
        then F.x = f1.x by A1,A3,Th12;
        hence y in f1.:A by A3,A4,FUNCT_2:35;
      end;
      let y be object;
      assume y in f1.:A;
      then consider x being Element of X1 such that
A5:   x in A & y = f1.x by FUNCT_2:65;
      x is Point of X by PRE_TOPC:25;
      then
A6:   F.x = f1.x by A1,Th12;
      x in the carrier of X1 union X2 by A2,XBOOLE_0:def 3;
      hence thesis by A5,A6,FUNCT_2:35;
    end;
  end;
  let A be Subset of X2;
  hereby
    let y be object;
    assume y in (f1 union f2).:A;
    then consider x being Element of X1 union X2 such that
A7: x in A and
A8: y = F.x by FUNCT_2:65;
    x is Point of X by PRE_TOPC:25;
    then F.x = f2.x by A1,A7,Th12;
    hence y in f2.:A by A7,A8,FUNCT_2:35;
  end;
  let y be object;
  assume y in f2.:A;
  then consider x being Element of X2 such that
A9: x in A & y = f2.x by FUNCT_2:65;
  x is Point of X by PRE_TOPC:25;
  then
A10: F.x = f2.x by A1,Th12;
  x in the carrier of X1 union X2 by A2,XBOOLE_0:def 3;
  hence thesis by A9,A10,FUNCT_2:35;
end;
