
theorem c1:
for R being add-associative right_zeroed right_complementable
            Abelian left-distributive non empty doubleLoopStr,
    a,b being Element of R,
    i being Integer holds i '*' (a * b) = (i '*' a) * b
proof
let R be add-associative right_zeroed right_complementable
         Abelian left-distributive non empty doubleLoopStr,
    a,b be Element of R,
    i be Integer;
defpred P[Integer] means $1'*'(a*b) = ($1'*'a)*b;
A2: P[0]
proof
  0 '*' a = 0.R by RING_3:59;
  hence 0'*'(a*b) = (0'*'a) * b by RING_3:59;
end;
A3: for u being Integer holds P[u] implies P[u - 1] & P[u + 1]
   proof
   let u be Integer;
   assume A4: P[u];
   thus P[u-1]
   proof
     set k = u-1;
A6:  (k+1)'*'(a*b) = (k'*'a+1'*'a)*b by A4,RING_3:62
       .= (k'*'a) * b + (1'*'a)*b by VECTSP_1:def 3
       .= (k'*'a) * b + a * b by RING_3:60;
     (k'*'a) * b + 0.R = (k'*'a) * b + (a * b + - a* b) by RLVECT_1:5
                      .= (k+1)'*'(a*b) + - a * b by A6,RLVECT_1:def 3;
     hence (k'*'a) * b = (k+1)'*'(a*b) + (-1) '*' (a * b) by RING_3:61
                      .= k '*' (a * b) by RING_3:62;
     end;
   thus P[u+1]
   proof
     set k = u+1;
     A6: (k-1)'*'(a*b) = (k'*'a-1'*'a)*b by A4,RING_3:64
                           .= (k'*'a) * b + (-(1'*'a))*b by VECTSP_1:def 3
                           .= (k'*'a) * b + -((1'*'a)*b) by VECTSP_1:9
                           .= (k'*'a) * b - a * b by RING_3:60;
     (k'*'a) * b + 0.R = (k'*'a) * b + (- a * b + a* b) by RLVECT_1:5
                      .= (k-1)'*'(a*b) + a * b by A6,RLVECT_1:def 3;
     hence (k'*'a) * b = (k-1)'*'(a*b) + 1 '*' (a * b) by RING_3:60
                      .= k '*' (a * b) by RING_3:62;
     end;
   end;
for i being Integer holds P[i] from INT_1:sch 4(A2,A3);
hence thesis;
end;
