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;
