reserve p,q for Rational;
reserve g,m,m1,m2,n,n1,n2 for Nat;
reserve i,i1,i2,j,j1,j2 for Integer;
reserve R for Ring, F for Field;

theorem Th64:
for R being add-associative right_zeroed right_complementable
            Abelian non empty doubleLoopStr,
    a being Element of R,
    i,j being Integer holds (i * j) '*' a = i '*' (j '*' a)
proof
let R be add-associative right_zeroed right_complementable
         Abelian non empty doubleLoopStr,
    a be Element of R, i,j be Integer;
defpred P[Integer] means
  for k being Integer st k = $1 holds (k*j) '*' a = k'*'(j '*' a);
now let k be Integer;
  assume A1: k = 0;
  hence (k*j) '*' a = 0.R by Th58 .= k'*'(j '*' a) by A1,Th58;
  end;
then A2: P[0];
A3: for u being Integer holds P[u] implies P[u - 1] & P[u + 1]
   proof
   let u be Integer;
   assume A4: P[u];
   now let k be Integer;
     assume A5: k = u-1;
     hence (k*j) '*' a = (u*j-j) '*' a
                      .= (u * j) '*' a - j '*' a by Th63
                      .= u '*' (j '*' a) - j '*' a by A4
                      .= k '*' (j '*' a) by A5,Lm6;
     end;
   hence P[u-1];
   now let k be Integer;
     assume A6: k = u+1;
     hence (k*j) '*' a = (u*j+j) '*' a
                      .= (u * j) '*' a + j '*' a by Th61
                      .= u '*' (j '*' a) + j '*' a by A4
                      .= k '*' (j '*' a) by A6,Lm5;
     end;
   hence P[u+1];
   end;
for i being Integer holds P[i] from INT_1:sch 4(A2,A3);
hence thesis;
end;
