reserve U1,U2,U3 for Universal_Algebra,
  n,m for Nat,
  x,y,z for object,
  A,B for non empty set,
  h1 for FinSequence of [:A,B:];
reserve h1 for homogeneous quasi_total non empty PartFunc of
    (the carrier of U1)*,the carrier of U1,
  h2 for homogeneous quasi_total non empty PartFunc of
    (the carrier of U2)*,the carrier of U2;

theorem Th15:
  for X being set, S being 1-sorted holds
    Carrier(X --> S) = X --> the carrier of S
proof
  let X be set, S be 1-sorted;
  A1: dom Carrier(X --> S) = dom(X --> S) by Def13
    .= dom(X --> the carrier of S);
  now
    let x be object;
    assume x in dom Carrier(X --> S); then
A2: x in dom(X --> S) by Def13;
    then consider R being 1-sorted such that
A4: R = (X-->S).x & (Carrier(X-->S)).x = the carrier of R
      by Def13;
    thus Carrier(X --> S).x = the carrier of S by A2, A4, FUNCOP_1:7
      .= (X --> the carrier of S).x by A2, FUNCOP_1:7;
  end;
  hence thesis by A1;
end;
