reserve a, I for set,
  S for non empty non void ManySortedSign;
reserve A, M for ManySortedSet of I,
  B, C for non-empty ManySortedSet of I;

theorem
  for A being non-empty MSAlgebra over S, C being MSCongruence of A for
  s being SortSymbol of S, x, y being Element of (the Sorts of A).s holds (
  MSNat_Hom(A,C)).s.x = (MSNat_Hom(A,C)).s.y iff [x,y] in C.s
proof
  let A be non-empty MSAlgebra over S, C be MSCongruence of A, s be SortSymbol
  of S, x, y be Element of (the Sorts of A).s;
  set f = (MSNat_Hom(A,C)).s, g = MSNat_Hom(A,C,s);
A1: f = g by MSUALG_4:def 16;
  hereby
    assume
A2: (MSNat_Hom(A,C)).s.x = (MSNat_Hom(A,C)).s.y;
    Class(C.s,x) = g.x by MSUALG_4:def 15
      .= Class(C.s,y) by A1,A2,MSUALG_4:def 15;
    hence [x,y] in C.s by EQREL_1:35;
  end;
  assume
A3: [x,y] in C.s;
  thus (MSNat_Hom(A,C)).s.x = Class(C.s,x) by A1,MSUALG_4:def 15
    .= Class(C.s,y) by A3,EQREL_1:35
    .= (MSNat_Hom(A,C)).s.y by A1,MSUALG_4:def 15;
end;
