reserve k,n,m for Nat,
  A,B,C for Ordinal,
  X for set,
  x,y,z for object;
reserve f,g,h,fx for Function,
  K,M,N for Cardinal,
  phi,psi for
  Ordinal-Sequence;
reserve a,b for Aleph;

theorem Th17:
  a c= M or a in M implies a +` M = M & M +` a = M & a *` M = M &
  M *` a = M
proof
 card 0 in a by Th15;
   then
A1: 0 in a;
  assume
A2: a c= M or a in M;
  then
A3: M is infinite by Th16;
A4: a c= M by A2,ORDINAL1:16;
  thus a +` M = M by CARD_2:76,A3,A4;
  thus M +` a = M by CARD_2:76,A3,A4;
  thus a *` M = M by A1,CARD_4:16,A3,A4;
  thus M *` a = M by A1,CARD_4:16,A3,A4;
end;
