 reserve o,o1,o2 for object;
 reserve n for Ordinal;
 reserve R,L for non degenerated comRing;
 reserve b for bag of 1;

theorem
   for b1,b2 be bag of 1 holds b1 + b2 = 1 --> ((b1.0) + (b2.0))
    proof
      let b1,b2 be bag of 1;
      reconsider b = b1 + b2 as bag of 1;
A1:   dom(b1 + b2) = {0} by Th4 .=  dom (1 --> ((b1.0) + (b2.0))) by Th4;
      for x being object st x in dom(b1 + b2) holds
      (b1 + b2).x = (1 --> ((b1.0) + (b2.0))).x
      proof
        let x be object;
        assume
A2:     x in dom(b1 + b2); then
        x in {0} by Th4; then
        (b1 + b2).x = (b1 + b2).0 by TARSKI:def 1
        .= (b1.0) + (b2.0) by PRE_POLY:def 5
        .= (1 --> ((b1.0) + (b2.0))).x by A2,FUNCOP_1:7;
        hence thesis;
      end;
      hence thesis by A1;
    end;
