theorem Th70:
  X is non-trivial & the_sort_of t = s implies
  card Coim(t,a) c= card Coim(C-sub t,a)
  proof assume that
ZZ: X is non-trivial and
Z0: the_sort_of t = s;
    defpred P[context of x] means
    for C st C = $1 holds card Coim(t,a) c= card Coim(C-sub t,a);
A0: P[x-term] by Z0,Th41;
A1: for o,p st p is x-context_including
    holds P[x-context_in p] implies
    for C being context of x st C = o-term p holds P[C]
    proof
      let o,p;
      assume Z1: p is x-context_including;
      assume Z2: P[x-context_in p];
      set i = x-context_pos_in p;
A6:   i in dom p = dom the_arity_of o by Z1,Th71,MSUALG_3:6;
      then consider j such that
A3:   i = 1+j by NAT_1:10,FINSEQ_3:25;
      card (<*j*>^^Coim((x-context_in p)-sub t,a))
      = card Coim((x-context_in p)-sub t,a) by Th1;
      then consider f being Function such that
A2:   f is one-to-one & dom f = Coim(t,a) &
      rng f c= <*j*>^^Coim((x-context_in p)-sub t,a)
      by Z2,CARD_1:10;
      let C be context of x;
      assume Z3: C = o-term p;
      x-context_in p = p.i by Z1,Th71;
      then x-context_in p in (the Sorts of Free(S,X)).((the_arity_of o)/.i)
      by A6,MSUALG_6:2;
      then the_sort_of (x-context_in p) = (the_arity_of o)/.i by SORT;
      then reconsider q = p+*(x-context_pos_in p,(x-context_in p)-sub t)
      as Element of Args(o,Free(S,X)) by MSUALG_6:7;
A4:   C-sub t = o-term q by ZZ,Z0,Z1,Z3,Th43;
      <*j*>^^Coim((x-context_in p)-sub t,a) c= Coim(C-sub t,a)
      proof
        let r be object;
        assume r in <*j*>^^Coim((x-context_in p)-sub t,a);
        then consider n being Element of Coim((x-context_in p)-sub t,a)
        such that
A5:     r = <*j*>^n & n in Coim((x-context_in p)-sub t,a);
        Coim((x-context_in p)-sub t,a) c= dom((x-context_in p)-sub t)
        by RELAT_1:132;
        then reconsider n as Element of dom((x-context_in p)-sub t) by A5;
        i <= len p & dom q = dom the_arity_of o by A6,FINSEQ_3:25,MSUALG_3:6;
        then j < len p = len q & q.i = (x-context_in p)-sub t
        by A3,Z1,Th71,MSUALG_3:6,NAT_1:13,FINSEQ_3:29,FUNCT_7:31;
        then
A9:     <*j*>^n in dom (C-sub t) &
        (C-sub t).r = ((x-context_in p)-sub t).n in {a}
        by A3,A4,A5,TREES_4:11,12,FUNCT_1:def 7;
        thus thesis by A5,A9,FUNCT_1:def 7;
      end;
      then rng f c= Coim(C-sub t,a) by A2;
      hence P[C] by A2,CARD_1:10;
    end;
    P[C] from ContextInd(A0,A1);
    hence thesis;
end;
