theorem Th61:
  for a being SortSymbol of S
  for x being Element of (the generators of G).a holds
  @x value_at(C,s) = s.a.x
  proof
    let a be SortSymbol of S;
    let x be Element of (the generators of G).a;
    s is ManySortedFunction of the generators of G, the Sorts of C
    by AOFA_A00:48;
    then consider h being ManySortedFunction of T,C such that
A1: h is_homomorphism T,C & s = h||the generators of G by AOFA_A00:def 19;
    @x value_at(C,s) = h.a.x by A1,Th29 .= ((h.a)|((the generators of G).a)).x
    by FUNCT_1:49;
    hence @x value_at(C,s) = s.a.x by A1,MSAFREE:def 1;
  end;
