
theorem qua4:
for L being non empty ZeroStr
for a,b,c being Element of L holds deg <%c,b,a%> = 2 iff a <> 0.L
proof
   let L be non empty ZeroStr; let a,b,c be Element of L;
H: deg <%c,b,a%> = len <%c,b,a%> - 1 by HURWITZ:def 2;
A: now assume a <> 0.L;
   then len <%c,b,a%> = 3 by qua3;
   hence deg <%c,b,a%> = 2 by H;
   end;
now assume deg <%c,b,a%> = 2;
   then len <%c,b,a%> = 2 + 1 by H;
   then <%c,b,a%>.2 <> 0.L by ALGSEQ_1:10;
   hence a <> 0.L by qua1;
   end;
hence thesis by A;
end;
