
theorem Lm0x:
for R being add-associative right_zeroed right_complementable
            Abelian non empty doubleLoopStr
for i being Integer holds i '*' 0.R = 0.R
proof
let R be add-associative right_zeroed right_complementable
            Abelian non empty doubleLoopStr, i be Integer;
defpred P[Integer] means ($1) '*' 0.R = 0.R;
A2: P[0] by RING_3:59;
A3: for u being Integer holds P[u] implies P[u - 1] & P[u + 1]
   proof
   let u be Integer;
   assume A4: P[u];
   set a = 0.R;
   now let k be Integer;
     assume k = u-1;
     hence k'*'a
         = u '*' a - 1 '*' a by RING_3:64
        .= 0.R - 0.R by A4,RING_3:60
        .= 0.R;
     end;
   hence P[u-1];
   now let k be Integer;
     assume k = u+1;
     hence k'*'a
         = u '*' a + 1 '*' a by RING_3:62
        .= 0.R by A4,RING_3:60;
     end;
   hence P[u+1];
   end;
for i being Integer holds P[i] from INT_1:sch 4(A2,A3);
hence thesis;
end;
