theorem
  f|Y is constant implies f|X|Y is constant
proof
  assume f|Y is constant;
  then consider d such that
A1: for c st c in Y /\ dom f holds f/.c = d by Th35;
  take d;
  let c;
  assume
A2: c in dom (f|X|Y);
  then
A3: c in Y /\ dom (f|X) by RELAT_1:61;
  then
A4: c in Y by XBOOLE_0:def 4;
A5: c in dom (f|X) by A3,XBOOLE_0:def 4;
  then c in dom f /\ X by RELAT_1:61;
  then c in dom f by XBOOLE_0:def 4;
  then c in Y /\ dom f by A4,XBOOLE_0:def 4;
  then f/.c = d by A1;
  then (f|X)/.c = d by A5,Th15;
  then (f|X|Y)/.c = d by A3,Th16;
  hence thesis by A2,PARTFUN1:def 6;
end;
