theorem Th1:
  x in (a+^b)\a iff ex c st x = a+^c & c in b
  proof
A1: x in (a+^b)\a iff a c= x & x in a+^b by ORDINAL6:5;
    hereby assume
A2:   x in (a+^b)\a; then
      reconsider c = x as Ordinal;
      take d = c-^a;
      thus x = a+^d by A1,A2,ORDINAL3:def 5;
      hence d in b by A2,ORDINAL3:22;
    end;
    given c such that
A3: x = a+^c & c in b;
    a c= x & x in a+^b by A3,ORDINAL2:32,ORDINAL3:24;
    hence thesis by ORDINAL6:5;
  end;
