reserve A for QC-alphabet;
reserve n,k,m for Nat;
reserve F,G,G9,H,H9 for Element of QC-WFF(A);

theorem Th3:
  k in dom list_of_immediate_constituents(F) & G = (
  list_of_immediate_constituents(F)).k implies G is_immediate_constituent_of F
proof
  assume that
A1: k in dom list_of_immediate_constituents(F) and
A2: G = (list_of_immediate_constituents(F)).k;
A3:  list_of_immediate_constituents(F) <> <*>QC-WFF(A) by A1;
A4: F <> VERUM(A) & not F is atomic by Def1,A3;
  per cases by A4,QC_LANG1:9;
  suppose
A5: F is negative;
    then
A6: list_of_immediate_constituents(F) = <* the_argument_of F *> by Def1;
    then k in Seg 1 by A1,FINSEQ_1:def 8;
    then k = 1 by FINSEQ_1:2,TARSKI:def 1;
    then G = the_argument_of F by A2,A6;
    hence thesis by A5,QC_LANG2:48;
  end;
  suppose
A7: F is universal;
    then
A8: not F is conjunctive by QC_LANG1:20;
    (not F is atomic)& not F is negative by A7,QC_LANG1:20;
    then
A9: list_of_immediate_constituents(F) = <* the_scope_of F *>
      by A8,Def1,A4;
    then k in Seg 1 by A1,FINSEQ_1:def 8;
    then k = 1 by FINSEQ_1:2,TARSKI:def 1;
    then G = the_scope_of F by A2,A9;
    hence thesis by A7,QC_LANG2:50;
  end;
  suppose
A10: F is conjunctive;
A13: list_of_immediate_constituents(F) = <* the_left_argument_of F,
    the_right_argument_of F *> by A10,Def1;
    len <* the_left_argument_of F, the_right_argument_of F *> = 2 by
FINSEQ_1:44;
    then
A14: k in {1,2} by A1,A13,FINSEQ_1:2,def 3;
    now
      per cases by A14,TARSKI:def 2;
      suppose
        k = 1;
        hence thesis by A2,A10,A13,QC_LANG2:49;
      end;
      suppose
        k = 2;
        hence thesis by A2,A10,A13,QC_LANG2:49;
      end;
    end;
    hence thesis;
  end;
  suppose
A15: F is universal;
    then
A16: not F is conjunctive by QC_LANG1:20;
    (not F is atomic)& not F is negative by A15,QC_LANG1:20;
    then
A17: list_of_immediate_constituents(F) = <* the_scope_of F *>
      by A16,Def1,A4;
    then k in Seg 1 by A1,FINSEQ_1:def 8;
    then k = 1 by FINSEQ_1:2,TARSKI:def 1;
    then G = the_scope_of F by A2,A17;
    hence thesis by A15,QC_LANG2:50;
  end;
end;
