reserve Al for QC-alphabet;
reserve a,b,c,d for object,
  i,k,n for Nat,
  p,q for Element of CQC-WFF(Al),
  x,y,y1 for bound_QC-variable of Al,
  A for non empty set,
  J for interpretation of Al,A,
  v,w for Element of Valuations_in(Al,A),
  f,g for Function,
  P,P9 for QC-pred_symbol of k,Al,
  ll,ll9 for CQC-variable_list of k,Al,
  l1 for FinSequence of QC-variables(Al),
  Sub,Sub9,Sub1 for CQC_Substitution of Al,
  S,S9,S1,S2 for Element of CQC-Sub-WFF(Al),
  s for QC-symbol of Al;
reserve vS,vS1,vS2 for Val_Sub of A,Al;
reserve B for Element of [:QC-Sub-WFF(Al),bound_QC-variables(Al):],
  SQ for second_Q_comp of B;
reserve B for CQC-WFF-like Element of [:QC-Sub-WFF(Al),
  bound_QC-variables(Al):],
  xSQ for second_Q_comp of [S,x],
  SQ for second_Q_comp of B;
reserve B1 for Element of [:QC-Sub-WFF(Al),bound_QC-variables(Al):];
reserve SQ1 for second_Q_comp of B1;
reserve a for Element of A;

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;
