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 Th76:
  s + t <=' r + t iff s <=' r
proof
  thus s + t <=' r + t implies s <=' r
  proof
    given z such that
A1: r+t = s+t+z;
    take z;
    r+t = s+z+t by A1,Th51;
    hence thesis by Th62;
  end;
  given z such that
A2: r = s+z;
  take z;
  thus thesis by A2,Th51;
end;
