reserve S for non empty non void ManySortedSign;
reserve X for non-empty ManySortedSet of S;
reserve x,y,z for set, i,j for Nat;

theorem Th35:
  for A being non-empty MSAlgebra over S
  for B being A-Image MSAlgebra over S
  for s being SortSymbol of S
  for e being Element of (Equations S).s
  st A |= e holds B |= e
  proof
    let A be non-empty MSAlgebra over S;
    let B be A-Image MSAlgebra over S;
    consider f being ManySortedFunction of A,B such that
A1: f is_epimorphism A,B by Def5;
    let s be SortSymbol of S;
    let e be Element of (Equations S).s;
    assume A2: A |= e;
    let h be ManySortedFunction of TermAlg S, B;
    assume
A3: h is_homomorphism TermAlg S, B;
    consider g being ManySortedFunction of TermAlg S,A such that
A4: g is_homomorphism TermAlg S, A & h = f**g by A1,A3,EQUATION:24;
    h.s = (f.s)*(g.s) & e`1 in (the Sorts of TermAlg S).s &
    e`2 in (the Sorts of TermAlg S).s
    by A4,MSUALG_3:2,EQUATION:29,30; then
    h.s.e`1 = f.s.(g.s.e`1) & h.s.e`2 = f.s.(g.s.e`2) by FUNCT_2:15;
    hence h.s.e`1 = h.s.e`2 by A2,A4;
  end;
