theorem Th10:
  u <> v & u in Lex(E) & v in Lex(E) implies not ex w st u^w = v or w^u = v
proof
  assume that
A1: u <> v and
A2: u in Lex(E) and
A3: v in Lex(E);
A4: len u = 1 by A2,Th9;
  given w such that
A5: u^w = v or w^u = v;
A6: len v = 1 by A3,Th9;
  per cases by A5;
  suppose
A7: u^w = v;
    len (u^w) = 1 + len w by A4,AFINSQ_1:17;
    then w = <%>E by A6,A7;
    hence contradiction by A1,A7;
  end;
  suppose
A8: w^u = v;
    len (w^u) = 1 + len w by A4,AFINSQ_1:17;
    then w = <%>E by A6,A8;
    hence contradiction by A1,A8;
  end;
end;
