theorem Th93:
  X is non-trivial & the_sort_of C = s1 implies
  for x1 being Element of X.s1
  for C1 being (context of x1), C2 being context of x st C2 = C1-sub(C)
  holds transl C2 = (transl C1)*(transl C)
  proof
    assume that
ZZ: X is non-trivial and
Z0: the_sort_of C = s1;
    let x1 be Element of X.s1;
    let C1 be (context of x1);
    let C2 be context of x;
    assume Z1: C2 = C1-sub(C);
    reconsider f = transl C as Function of (the Sorts of Free(S,X)).s,
    (the Sorts of Free(S,X)).s1 by Z0;
    transl C2 = (transl C1)*f
    proof
      let t be Element of (the Sorts of Free(S,X)).s;
A1:   the_sort_of t = s by SORT;
      then
A2:   (transl C2).t = C2-sub t & (transl C).t = C-sub t by TRANS;
      the_sort_of (C-sub t) = s1 by Z0,SORT;
      then (transl C1).(C-sub t) = C1-sub(C-sub t) by TRANS;
      then ((transl C1)*f).t = C1-sub(C-sub t) by A2,FUNCT_2:15;
      hence thesis by ZZ,A2,Z0,Z1,A1,Th60;
    end;
    hence transl C2 = (transl C1)*(transl C);
  end;
