reserve k,m,n for Element of NAT,
  a,X,Y for set,
  D,D1,D2 for non empty set;
reserve p,q for FinSequence of NAT;
reserve x,y,z,t for Variable;
reserve F,F1,G,G1,H,H1 for ZF-formula;
reserve sq,sq9 for FinSequence;

theorem Th54:
  F is_immediate_constituent_of All(x,H) iff F = H
proof
  thus F is_immediate_constituent_of All(x,H) implies F = H
  proof
A1: now
      given G such that
A2:   All(x,H) = F '&' G or All(x,H) = G '&' F;
      (F '&' G).1 = 3 & (G '&' F).1 = 3 by Th16;
      hence contradiction by A2,Th17;
    end;
A3: now
      assume
A4:   All(x,H) = 'not' F;
      All(x,H).1 = 4 by Th17;
      hence contradiction by A4,FINSEQ_1:41;
    end;
    assume F is_immediate_constituent_of All(x,H);
    then ex y st All(x,H) = All(y,F) by A3,A1;
    hence thesis by Th3;
  end;
  thus thesis;
end;
