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 Th61:
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 '*' a + 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 (i + k)'*'a = i'*'a + k'*'a;
now let k be Integer;
  assume A1: k = 0;
  hence (i + k)'*'a = i'*'a + 0.R .= i'*'a + k'*'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 k = u-1;
     then A5: (i + (k+1))'*'a = i'*'a + (k+1)'*'a by A4
                             .= i'*'a + (k'*'a + a) by Lm5
                             .= (i'*'a + k'*'a) + a by RLVECT_1:def 3;
     (i + (k+1))'*'a = ((i + k)+1)'*'a
                    .= (i+k)'*'a  + a by Lm5;
     hence (i + k)'*'a = i'*'a + k'*'a by A5,RLVECT_1:8;
     end;
   hence P[u-1];
   now let k be Integer;
     assume k = u+1;
     then A6: (i + (k-1))'*'a = i'*'a + (k-1)'*'a by A4
                             .= i'*'a + (k'*'a - a) by Lm6
                             .= (i'*'a + k'*'a) - a by RLVECT_1:def 3;
     (i + (k-1))'*'a = ((i + k)-1)'*'a
                    .= (i+k)'*'a  - a by Lm6;
     hence (i + k)'*'a = i'*'a + k'*'a by A6,RLVECT_1:8;
     end;
   hence P[u+1];
   end;
 for i being Integer holds P[i] from INT_1:sch 4(A2,A3);
hence thesis;
end;
