theorem
   x1 <> x2 & t is x2-omitting implies  Hom(Free(S,X),x1,x2).t is x1-omitting
  proof
    assume
A0: x1 <> x2;
    set T = Free(S,X);
    set h = Hom(T,x1,x2);
    set s0 = s;
    defpred P[Element of T] means
    $1 is x2-omitting implies h.$1 is x1-omitting;
A1: for s,x holds P[x-term]
    proof
      let s,x;
      set r = x-term;
      assume Z1: r is x2-omitting;
      per cases;
      suppose
A2:     s0 <> s or x1 <> x & x2 <> x;
        then Hom(T,x1,x2).s.r = r & the_sort_of @r = the_sort_of r
        by HOM;
        then Hom(T,x1,x2).(the_sort_of r).r = r & x-term is x1-omitting
        by A2,SORT,ThC1;
        hence h.r is x1-omitting by ABBR;
      end;
      suppose
        s0 = s & x1 = x;
        then h.s.r = x2-term & the_sort_of @r = the_sort_of r by HOM;
        then
A4:     h.(the_sort_of r).r = x2-term by SORT;
        x2-term is x1-omitting by A0,ThC1;
        hence h.r is x1-omitting by A4,ABBR;
      end;
      suppose
A3:     s0 = s & x2 = x;
        thus h.r is x1-omitting by Z1,A3;
      end;
    end;
A2: for o, p st for t being Element of T st t in rng p holds P[t]
    holds P[o-term p]
    proof
      let o, p;
      set r = o-term p;
      assume Z2: for t being Element of T st t in rng p holds P[t];
      assume
S4:   r is x2-omitting;
A6:   the_sort_of r = the_result_sort_of o by Th8;
      reconsider q = p as Element of Args(o,T);
      reconsider m = h#q as Element of Args(o,Free(S,X));
A7:   Den(o,T).q = Den(o,Free(S,X)).p = o-term p by MSAFREE4:13;
A9:   h.(the_result_sort_of o).r = Den(o,Free(S,X)).m
      by A7,MSUALG_3:def 7,MSUALG_6:def 2
      .= o-term m by MSAFREE4:13;
      now let i; assume
B1:     i in dom m;
B2:     dom m = dom the_arity_of o = dom p by MSUALG_3:6;
        then
B3:     p.i in rng p & p/.i = p.i = q/.i by B1,FUNCT_1:def 3,PARTFUN1:def 6;
        q/.i in (the Sorts of T).((the_arity_of o)/.i) by B1,B2,B3,MSUALG_6:2;
        then
B5:     the_sort_of (q/.i) = (the_arity_of o)/.i by SORT;
        m/.i = m.i = h.((the_arity_of o)/.i).(q/.i)
        by B1,PARTFUN1:def 6,B2,B3,MSUALG_3:def 6;
        then h.(q/.i) = m/.i & p/.i is x2-omitting by B1,B2,S4,B5,ABBR,Th54;
        hence m/.i is x1-omitting by B3,Z2;
      end;
      then o-term m is x1-omitting by Th54;
      hence h.r is x1-omitting by A6,A9,ABBR;
    end;
    thus P[t] from TermInd(A1,A2);
  end;
