theorem Th48:
  card I in dom stop I & (stop I).card I = halt S
proof
A1: (Stop S). 0 = halt S;
A2: 0 in dom Stop S by Th2;
  set pI=stop I;
  card pI=card I+1 by Th39;
  then card I <card pI by XREAL_1:29;
  hence card I in dom pI by AFINSQ_1:66;
  pI.(0 qua Nat +card I) = halt S by A1,A2,AFINSQ_1:def 3;
  hence thesis;
end;
