reserve f,g,h for Function,
  A for set;
reserve F for Function,
  B,x,y,y1,y2,z for set;
reserve x,z for object;

theorem Th12:
  (A --> x)|B = A /\ B --> x
proof
A1: A = {} or A <> {};
A2: A /\ B = {} or A /\ B <> {};
A3: dom ((A --> x)|B) = dom (A --> x) /\ B by RELAT_1:61
    .= A /\ B by A1,RELAT_1:160
    .= dom (A /\ B --> x) by A2,RELAT_1:160;
  now
    let z be object such that
A4: z in dom (A /\ B --> x);
    A /\ B = {} or A /\ B <> {};
    then
A5: z in A /\ B by A4,RELAT_1:160;
    then
A6: z in A by XBOOLE_0:def 4;
    thus ((A --> x)|B).z = (A --> x).z by A3,A4,FUNCT_1:47
      .= x by A6,Th7
      .= (A /\ B --> x).z by A5,Th7;
  end;
  hence thesis by A3;
end;
