theorem Th58:
  C9 = o-term k & z-context_in k = z-term & k1 = k+*(z-context_pos_in k,l)
  implies C9-sub l = o-term k1
  proof set C = C9, p = k, p1 = k1, x = z, t = l, X = Z;
    assume Z0: C = o-term p;
    assume Z1: x-context_in p = x-term;
    assume Z2: p1 = p+*(x-context_pos_in p,t);
    set i = x-context_pos_in p;
A1: p is x-context_including by Z0,Th53;
    then
A2: i in dom p = dom the_arity_of o & p.i = x-term by Z1,Th71,MSUALG_3:6;
    p1.i = t by A1,Z2,Th71,FUNCT_7:31;
    then t in (the Sorts of Free(S,X)).((the_arity_of o)/.i) &
    x-term in (the Sorts of Free(S,X)).((the_arity_of o)/.i) by A2,MSUALG_6:2;
    then
A3: the_sort_of t = (the_arity_of o)/.i = the_sort_of (x-context_in p) = s
    by Z1,SORT;
    then (x-context_in p)-sub t = t by Z1,Th41;
    hence C-sub t = o-term p1 by Z0,Z2,A3,Th43,Th53;
  end;
