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|X is constant & Y c= X implies f|Y is constant
proof
  assume that
A1: f|X is constant and
A2: Y c= X;
  consider d such that
A3: for c st c in X /\ dom f holds f/.c = d by A1,Th35;
  now
    let c;
    assume c in Y /\ dom f;
    then c in Y & c in dom f by XBOOLE_0:def 4;
    then c in X /\ dom f by A2,XBOOLE_0:def 4;
    hence f/.c = d by A3;
  end;
  hence thesis by Th35;
end;
