reserve A,B,C for Ordinal;
reserve a,b,c,d for natural Ordinal;
reserve l,m,n for natural Ordinal;
reserve i,j,k for Element of omega;
reserve x,y,z for Element of RAT+;
reserve i,j,k for natural Ordinal;

theorem Th58:
  for i,j being ordinal Element of RAT+ holds i+j = i+^j
proof
  let i,j be ordinal Element of RAT+;
  set ni = numerator i, nj = numerator j;
A1: j in omega by ORDINAL1:def 12;
  then
A2: denominator j = 1 by Def9;
A3: i in omega by ORDINAL1:def 12;
  then denominator i = 1 by Def9;
  hence i+j = (ni*^1+^1*^nj)/1 by A2,ORDINAL2:39
    .= (ni+^1*^nj)/1 by ORDINAL2:39
    .= (ni+^nj)/1 by ORDINAL2:39
    .= ni+^nj by Th40
    .= i+^nj by A3,Def8
    .= i+^j by A1,Def8;
end;
