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 Th53:
  F is_immediate_constituent_of G '&' H iff F = G or F = H
proof
  thus F is_immediate_constituent_of G '&' H implies F = G or F = H
  proof
A1: now
      given x such that
A2:   G '&' H = All(x,F);
      (G '&' H).1 = 3 by Th16;
      hence contradiction by A2,Th17;
    end;
A3: now
      assume
A4:   G '&' H = 'not' F;
      (G '&' H).1 = 3 by Th16;
      hence contradiction by A4,FINSEQ_1:41;
    end;
    assume F is_immediate_constituent_of G '&' H;
    then ex H1 st G '&' H = F '&' H1 or G '&' H = H1 '&' F by A3,A1;
    hence thesis by Th30;
  end;
  assume F = G or F = H;
  hence thesis;
end;
