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
  c in dom f1 & f1 = f \/ g implies f1/.c = f/.c or f1/.c = g/.c
proof
  assume that
A1: c in dom f1 and
A2: f1 = f \/ g;
  [c,f1/.c] in f1 by A1,Th46;
  then
A3: [c,f1/.c] in f or [c,f1/.c] in g by A2,XBOOLE_0:def 3;
  now
    per cases by A3,FUNCT_1:1;
    suppose
      c in dom f;
      then [c,f/.c] in f by Th46;
      then [c,f/.c] in f1 by A2,XBOOLE_0:def 3;
      hence thesis by Th46;
    end;
    suppose
      c in dom g;
      then [c,g/.c] in g by Th46;
      then [c,g/.c] in f1 by A2,XBOOLE_0:def 3;
      hence thesis by Th46;
    end;
  end;
  hence thesis;
end;
