theorem
  not x==0_No & not y == 0_No implies (x*y)" == x" * y"
proof
  assume
A1: not x==0_No & not y == 0_No; then
A2: x*x" == 1_No == y*y" by Th33;
  (x" * y") * (x*y) == (x" * y") * y*x == (x" * (y" * y))*x by SURREALR:54,69;
  then (x" * y") * (x*y) == (y" * y)*x"*x == (y" * y)*(x"*x)
  by SURREALO:4,SURREALR:69;
  then (x" * y") * (x*y) == (y" * y)*(x"*x) == (y" * y)*1_No
  by A2,SURREALO:4,SURREALR:54;
  then (x" * y") * (x*y) == y" * y by SURREALO:4;
  then (x" * y") * (x*y) == 1_No by A2,SURREALO:4;
  hence thesis by A1,Lm3,Th41;
end;
