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|SC = dom (f|SC) --> d implies f|SC is constant
proof
  assume
A1: f|SC = dom (f|SC) --> d;
  now
    let c;
    assume c in SC /\ dom f;
    then
A2: c in dom (f|SC) by RELAT_1:61;
    then f|SC/.c = d by A1,Th29;
    hence f/.c = d by A2,Th15;
  end;
  hence thesis by Th35;
end;
