
theorem Th2:
  addcomplex||REAL = addreal
  proof
    set ad = addcomplex||REAL;
    [:REAL,REAL:] c= [:COMPLEX,COMPLEX:] by NUMBERS:11,ZFMISC_1:96;
    then
A1: [:REAL,REAL:] c= dom(addcomplex) by FUNCT_2:def 1;
    then
A2: dom ad = [:REAL,REAL:] by RELAT_1:62;
A3: dom(addreal) = [:REAL,REAL:] by FUNCT_2:def 1;
    for z be object st z in dom ad holds ad.z = addreal.z
    proof
      let z be object;
      assume
A4:   z in dom ad;
      then consider x, y be object such that
A5:   x in REAL & y in REAL & z = [x,y] by A2,ZFMISC_1:def 2;
      reconsider x1 = x, y1 = y as Real by A5;
      thus ad.z = addcomplex.(x1,y1) by A4,A5,A2,FUNCT_1:49
      .= x1+y1 by BINOP_2:def 3
      .= addreal.(x1,y1) by BINOP_2:def 9
      .= addreal.z by A5;
    end;
    hence thesis by A1,A3,RELAT_1:62;
  end;
