theorem Th30:
  max+(R_EAL f) = max+f & max-(R_EAL f) = max-f
proof
A1: dom max+(R_EAL f) = dom R_EAL f by MESFUNC2:def 2
    .= dom max+f by RFUNCT_3:def 10;
  now
    let x be object;
    assume
A2: x in dom max+(R_EAL f);
    then (max+(R_EAL f)).x = max+(f.x) by MESFUNC2:def 2;
    hence (max+(R_EAL f)).x = (max+f).x by A1,A2,RFUNCT_3:def 10;
  end;
  hence max+(R_EAL f) = max+f by A1,FUNCT_1:2;
A3: dom max-(R_EAL f) = dom R_EAL f by MESFUNC2:def 3
    .=dom max-f by RFUNCT_3:def 11;
  now
    let x be object;
    assume
A4: x in dom max-(R_EAL f);
    (max-(R_EAL f)).x = max(-(((R_EAL f).x)),0.) by MESFUNC2:def 3,A4;
    then (max-(R_EAL f)).x = max-(f.x) by SUPINF_2:2;
    hence (max-(R_EAL f)).x = (max-f).x by A3,A4,RFUNCT_3:def 11;
  end;
  hence thesis by A3,FUNCT_1:2;
end;
