theorem Th57:
  rng l1 c= bound_QC-variables(Al) implies still_not-bound_in l1 = rng l1
proof
A1: variables_in(l1,bound_QC-variables(Al)) = { l1.k : 1 <= k & k <= len l1 &
   l1.k in bound_QC-variables(Al)} by QC_LANG3:def 1;
  assume
A2: rng l1 c= bound_QC-variables(Al);
A3: rng l1 c= variables_in(l1,bound_QC-variables(Al))
  proof
    let b be object;
    assume
A4: b in rng l1;
    then consider k being Nat such that
A5: k in dom l1 and
A6: l1.k = b by FINSEQ_2:10;
    k in Seg len l1 by A5,FINSEQ_1:def 3;
    then 1 <= k & k <= len l1 by FINSEQ_1:1;
    hence thesis by A2,A1,A4,A6;
  end;
  variables_in(l1,bound_QC-variables(Al)) c= rng l1
  proof
    let b be object;
    assume b in variables_in(l1,bound_QC-variables(Al));
    then consider k such that
A7: b = l1.k and
A8: 1 <= k & k <= len l1 and
    l1.k in bound_QC-variables(Al) by A1;
    k in Seg len l1 by A8,FINSEQ_1:1;
    then k in dom l1 by FINSEQ_1:def 3;
    hence thesis by A7,FUNCT_1:3;
  end;
  then variables_in(l1,bound_QC-variables(Al)) = rng l1 by A3;
  hence thesis by QC_LANG3:2;
end;
