reserve fi,psi for Ordinal-Sequence,
  A,A1,B,C,D for Ordinal,
  X,Y for set,
  x,y for object;

theorem
  A in B & C <> {} implies A in B*^C & A in C*^B
proof
  assume that
A1: A in B and
A2: C <> {};
  {} c= C;
  then {} c< C by A2;
  then {} in C by ORDINAL1:11;
  then
A3: 1 c= C by Lm1,ORDINAL1:21;
  then
A4: 1*^B c= C*^B by ORDINAL2:41;
A5: 1*^B = B by ORDINAL2:39;
A6: B*^1 = B by ORDINAL2:39;
  B*^1 c= B*^C by A3,ORDINAL2:42;
  hence thesis by A1,A4,A6,A5;
end;
