
theorem Th23:
  for S1,S2 being non void non empty ManySortedSign for f,g being
  Function st f,g form_morphism_between S1,S2 for A being MSAlgebra over S2 for
o1 being OperSymbol of S1, o2 being OperSymbol of S2 st o2 = g.o1 holds Den(o1,
  A|(S1,f,g)) = Den(o2,A)
proof
  let S1,S2 be non void non empty ManySortedSign;
  let f,g be Function;
  assume
A1: f,g form_morphism_between S1,S2;
  then reconsider
  g9 = g as Function of the carrier' of S1, the carrier' of S2 by Th9;
  let A be MSAlgebra over S2;
  let o1 be OperSymbol of S1, o2 be OperSymbol of S2;
  assume o2 = g.o1;
  then (the Charact of A).o2 = ((the Charact of A)*g9).o1 by FUNCT_2:15
    .= (the Charact of A|(S1,f,g)).o1 by A1,Def3;
  hence thesis;
end;
