reserve U1,U2,U3 for Universal_Algebra,
  m,n for Nat,
  a for set,
  A for non empty set,
  h for Function of U1,U2;

theorem
  MSAlg (id the carrier of U1) = (id the Sorts of MSAlg U1)
proof
  set f = (id the Sorts of MSAlg U1);
  set h = id the carrier of U1;
A1: the carrier of MSSign U1 = {0} by MSUALG_1:def 8;
  then reconsider Z1 = the Sorts of MSAlg U1 as ManySortedSet of {0};
A2: now
    let i be set;
    MSAlg U1 = MSAlgebra(#MSSorts U1,MSCharact U1#) by MSUALG_1:def 11;
    then
A3: Z1.0 = (0 .--> the carrier of U1).0 by MSUALG_1:def 9
      .= the carrier of U1 by FUNCOP_1:72;
    assume
A4: i in {0};
    then i = 0 by TARSKI:def 1;
    hence f.0 = h by A1,A4,A3,MSUALG_3:def 1;
  end;
  MSAlg h = 0 .--> h by Def3;
  then
A5: (MSAlg h).0 = h by FUNCOP_1:72;
  now
    let a be object;
    assume
A6: a in {0};
    then a = 0 by TARSKI:def 1;
    hence f.a = (MSAlg h).a by A2,A5,A6;
  end;
  hence f = MSAlg h by A1,PBOOLE:3;
end;
