theorem Th44:
  R_EAL(abs f) =|.R_EAL f.|
proof
A1: dom R_EAL(abs f) = dom f by VALUED_1:def 11
    .= dom(|. R_EAL f .|) by MESFUNC1:def 10;
  now
    let x be object;
     reconsider fx=f.x as Real;
    (R_EAL(abs f)).x = |.fx qua Complex.| by VALUED_1:18;
    then
A2: (R_EAL(abs f)).x = |.f.x.| by Th43;
    assume x in dom R_EAL(abs f);
    hence (R_EAL(abs f)).x =( |. R_EAL f .| ).x by A1,A2,MESFUNC1:def 10;
  end;
  hence thesis by A1,FUNCT_1:2;
end;
