reserve x,y,X,Y for set;
reserve C,D,E for non empty set;
reserve SC for Subset of C;
reserve SD for Subset of D;
reserve SE for Subset of E;
reserve c,c1,c2 for Element of C;
reserve d,d1,d2 for Element of D;
reserve e for Element of E;
reserve f,f1,g for PartFunc of C,D;
reserve t for PartFunc of D,C;
reserve s for PartFunc of D,E;
reserve h for PartFunc of C,E;
reserve F for PartFunc of D,D;

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;
