theorem Th91:
  for s1,s2 being SortSymbol of S, V being finite set holds
  m in dom q & s1 = (the_arity_of o)/.m implies
  ex y being (Element of Y.s1), C being (context of y), q1 st
  y nin V & q1 = q+*(m,y-term) &
  q1 is y-context_including & y-context_in q1 = y-term & C = o-term q1 &
  m = y-context_pos_in q1 & transl C = transl(o,m,q,Free(S,Y))
  proof
    let s1,s2 be SortSymbol of S, V be finite set;
    assume Z1: m in dom q;
    assume Z2: s1 = (the_arity_of o)/.m;
    consider y being (Element of Y.s1), C being (context of y), q1 such that
A1: y nin V & C = o-term q1 & q1 = q+*(m,y-term) &
    q1 is y-context_including &
    m = y-context_pos_in q1 & y-context_in q1 = y-term by Z1,Z2,Th59;
    take y, C, q1;
    thus y nin V by A1;
    thus q1 = q+*(m,y-term) by A1;
    thus q1 is y-context_including by A1;
    thus y-context_in q1 = y-term by A1;
    thus C = o-term q1 by A1;
    thus m = y-context_pos_in q1 by A1;
    dom transl C = (the Sorts of Free(S,Y)).s1 by FUNCT_2:def 1;
    hence dom transl C = dom transl(o,m,q,Free(S,Y)) by Z2,MSUALG_6:def 4;
    let c be object; assume
    c in dom transl C;
    then reconsider c as Element of (the Sorts of Free(S,Y)).s1;
    reconsider q2 = q+*(m,c) as Element of Args(o,Free(S,Y))
    by Z2,MSUALG_6:7;
A6: transl(o,m,q,Free(S,Y)).c = Den(o,Free(S,Y)).q2
    by Z2,MSUALG_6:def 4
    .= o-term q2 by MSAFREE4:13;
    q2 = q1+*(m,c) by A1,FUNCT_7:34;
    then C-sub c = o-term q2 & the_sort_of c = s1 by A1,Th58,SORT;
    hence thesis by A6,TRANS;
  end;
