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;
reserve r,s,t for Element of RAT+;

theorem Th62:
  s + t = r + t implies s = r
proof
  assume
A1: s + t = r + t;
  set r1 = numerator r, r2 = denominator r;
  set t1 = numerator t, t2 = denominator t;
  set s1 = numerator s, s2 = denominator s;
A2: t2 <> {} by Th35;
A3: s2 <> {} by Th35;
  then
A4: s2*^t2 <> {} by A2,ORDINAL3:31;
A5: r2 <> {} by Th35;
  then r2*^t2 <> {} by A2,ORDINAL3:31;
  then (s1*^t2+^s2*^t1)*^(r2*^t2) = (r1*^t2+^r2*^t1)*^(s2*^t2) by A1,A4,Th45
    .= (r1*^t2+^r2*^t1)*^s2*^t2 by ORDINAL3:50;
  then (s1*^t2+^s2*^t1)*^r2*^t2 = (r1*^t2+^r2*^t1)*^s2*^t2 by ORDINAL3:50;
  then (s1*^t2+^s2*^t1)*^r2 = (r1*^t2+^r2*^t1)*^s2 by A2,ORDINAL3:33
    .= r1*^t2*^s2+^r2*^t1*^s2 by ORDINAL3:46;
  then s1*^t2*^r2+^s2*^t1*^r2 = r1*^t2*^s2+^r2*^t1*^s2 by ORDINAL3:46
    .= r1*^t2*^s2+^s2*^t1*^r2 by ORDINAL3:50;
  then s1*^t2*^r2 = r1*^t2*^s2 by ORDINAL3:21
    .= r1*^s2*^t2 by ORDINAL3:50;
  then s1*^r2*^t2 = r1*^s2*^t2 by ORDINAL3:50;
  then s1*^r2 = r1*^s2 by A2,ORDINAL3:33;
  then s1/s2 = r1/r2 by A3,A5,Th45
    .= r by Th39;
  hence thesis by Th39;
end;
