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

theorem Th5:
  <:f,g:>|A = <:f|A,g:>
proof
A1: dom (<:f,g:>|A) = dom <:f,g:> /\ A by RELAT_1:61
    .= dom f /\ dom g /\ A by FUNCT_3:def 7
    .= dom f /\ A /\ dom g by XBOOLE_1:16
    .= dom (f|A) /\ dom g by RELAT_1:61;
  now
A2: dom (<:f,g:>|A) c= dom <:f,g:> by RELAT_1:60;
    let x be object such that
A3: x in dom (<:f,g:>|A);
A4: x in dom (f|A) by A1,A3,XBOOLE_0:def 4;
    thus (<:f,g:>|A).x = <:f,g:>.x by A3,FUNCT_1:47
      .= [f.x, g.x] by A3,A2,FUNCT_3:def 7
      .= [(f|A).x, g.x] by A4,FUNCT_1:47;
  end;
  hence thesis by A1,FUNCT_3:def 7;
end;
