reserve a for set,
  i for Nat;
reserve MS for segmental non void 1-element ManySortedSign,
  A for non-empty MSAlgebra over MS;

theorem Th24:
  for A,B being non-empty MSAlgebra over MS st 1-Alg A = 1-Alg B
  holds the MSAlgebra of A = the MSAlgebra of B
proof
  let A,B be non-empty MSAlgebra over MS such that
A1: 1-Alg A = 1-Alg B;
A2: 1-Alg B = UAStr(#the_sort_of B, the_charact_of B#) by MSUALG_1:def 14;
A3: 1-Alg A = UAStr(#the_sort_of A, the_charact_of A#) by MSUALG_1:def 14;
A4: now
    let i be object such that
A5: i in the carrier of MS;
A6: ex c being Component of the Sorts of B st (the Sorts of B).i = c
    proof
      reconsider c = (the Sorts of B).i as Component of the Sorts of B by A5,
PBOOLE:139;
      take c;
      thus thesis;
    end;
    ex c being Component of the Sorts of A st (the Sorts of A).i = c
    proof
      reconsider c = (the Sorts of A).i as Component of the Sorts of A by A5,
PBOOLE:139;
      take c;
      thus thesis;
    end;
    then (the Sorts of A).i = the_sort_of B by A1,A3,A2,MSUALG_1:def 12
      .= (the Sorts of B).i by A6,MSUALG_1:def 12;
    hence (the Sorts of A).i = (the Sorts of B).i;
  end;
  the Charact of A = the charact of 1-Alg A by A3,MSUALG_1:def 13
    .= the Charact of B by A1,A2,MSUALG_1:def 13;
  hence thesis by A4,PBOOLE:3;
end;
