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 Th66:
for R being add-associative right_zeroed right_complementable
            Abelian left_unital distributive non empty doubleLoopStr,
    i,j being Integer holds (i * j) '*' 1.R = (i '*' 1.R) * (j '*' 1.R)
proof
let R be add-associative right_zeroed right_complementable
         Abelian left_unital distributive non empty doubleLoopStr,
    i,j be Integer;
defpred P[Integer] means
  for k being Integer st k = $1 holds
            (k * j) '*' 1.R = (k '*' 1.R) * (j '*' 1.R);
now let k be Integer;
  assume A1: k = 0;
  hence (k * j) '*' 1.R = 0.R * (j '*' 1.R) by Th58
                       .= (k '*' 1.R) * (j '*' 1.R) 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) '*' 1.R = (u*j-j) '*' 1.R
       .= (u*j) '*' 1.R - (j '*' 1.R) by Th63
       .= (u '*' 1.R) * (j '*' 1.R) - (j '*' 1.R) by A4
       .= ((u '*' 1.R) * (j '*' 1.R)) + -(1.R * (j'*'1.R))
       .= ((u '*' 1.R) * (j '*' 1.R)) + (-(1.R)) * (j '*' 1.R) by VECTSP_1:9
       .= ((u '*' 1.R) + -1.R) * (j '*' 1.R) by VECTSP_1:def 3
       .= ((u '*' 1.R) - (1 '*' 1.R)) * (j '*' 1.R) by Th59
       .= (k '*' 1.R) * (j '*' 1.R) by Th63,A5;
     end;
   hence P[u-1];
   now let k be Integer;
     assume A6: k = u+1;
     hence (k * j) '*' 1.R = (u*j+j) '*' 1.R
       .= (u*j) '*' 1.R + (j '*' 1.R) by Th61
       .= (u '*' 1.R) * (j '*' 1.R) + (j '*' 1.R) by A4
       .= ((u '*' 1.R) * (j '*' 1.R)) + (1.R * (j '*' 1.R))
       .= ((u '*' 1.R) + 1.R) * (j '*' 1.R) by VECTSP_1:def 3
       .= ((u '*' 1.R) + (1 '*' 1.R)) * (j '*' 1.R) by Th59
       .= (k '*' 1.R) * (j '*' 1.R) by Th61,A6;
     end;
   hence P[u+1];
   end;
for i being Integer holds P[i] from INT_1:sch 4(A2,A3);
hence thesis;
end;
