
theorem c1a:
for R being add-associative right_zeroed right_complementable
            Abelian left-distributive non empty doubleLoopStr,
    a being Element of R,
    i being Integer holds i '*' (-a) = -(i '*' a)
proof
let R be add-associative right_zeroed right_complementable
         Abelian left-distributive non empty doubleLoopStr,
    a be Element of R,
    i be Integer;
defpred P[Integer] means $1'*'(-a) = -($1'*'a);
A2: P[0]
    proof
      0 '*' (-a) = -0.R by RING_3:59 .= -(0'*'a) by RING_3:59;
      hence 0'*'(-a) = -(0'*'a);
    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) = -(k'*'a + 1'*'a) by A4,RING_3:62
                     .= -((k'*'a) + a) by RING_3:60
                     .= -(k'*'a) + -a by RLVECT_1:31;
     thus -(k'*'a) = -(k'*'a) + 0.R
                  .= -(k'*'a) + (-a + a) by RLVECT_1:5
                  .= ((k+1)'*'(-a)) + a by A6,RLVECT_1:def 3
                  .= (k'*'(-a) + 1'*'(-a)) + a by RING_3:62
                  .= (k'*'(-a) + (-a)) + a by RING_3:60
                  .= k'*'(-a) + ((-a) + a) by RLVECT_1:def 3
                  .= k'*'(-a) + 0.R by RLVECT_1:5
                  .= k'*'(-a);
     end;
   thus P[u+1]
   proof
     set k = u+1;
     A6: (k-1)'*'(-a) = -(k'*'a - 1'*'a) by A4,RING_3:64
                     .= -((k'*'a) - a) by RING_3:60
                     .= -(k'*'a) + --a by RLVECT_1:33;
     thus -(k'*'a) = -(k'*'a) + 0.R
                  .= -(k'*'a) + (a + -a) by RLVECT_1:5
                  .= ((k-1)'*'(-a)) + -a by A6,RLVECT_1:def 3
                  .= (k'*'(-a) - 1'*'(-a)) + -a by RING_3:64
                  .= (k'*'(-a) - (-a)) + -a by RING_3:60
                  .= k'*'(-a) + ((-a) + a) by RLVECT_1:def 3
                  .= k'*'(-a) + 0.R by RLVECT_1:5
                  .= k'*'(-a);
     end;
   end;
for i being Integer holds P[i] from INT_1:sch 4(A2,A3);
hence thesis;
end;
