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+;

theorem Th42:
  b <> {} implies numerator (a/b) = RED(a,b) & denominator (a/b) = RED(b,a)
proof
  assume
A1: b <> {};
  per cases;
  suppose
A2: RED(b,a) = 1;
    then a/b = RED(a,b) by Def10;
    hence thesis by A2,Def8,Def9;
  end;
  suppose
A3: RED(b,a) <> 1;
A4: not [RED(a,b), RED(b,a)] in omega by Th32;
    a/b = [RED(a,b), RED(b,a)] by A1,A3,Def10;
    hence thesis by A4,Def8,Def9;
  end;
end;
