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 Th62:
for R being add-associative right_zeroed right_complementable
            Abelian 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 non empty doubleLoopStr,
    a be Element of R, i be Integer;
defpred P[Integer] means
  for k being Integer st k = $1 holds (-k) '*' a = -(k '*' a);
now let k be Integer;
  assume A1: k = 0;
  hence (-k)'*'a = -0.R by Th58 .= -(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 A5: k = u-1;
     hence (-k) '*' a= ((0-u)+1)'*'a
                    .= (0-u) '*' a + a by Lm5
                    .= -(u '*' a) + a by A4
                    .= -(u '*' a - a) by RLVECT_1:33
                    .= -(k '*' a) by Lm6,A5;
     end;
   hence P[u-1];
   now let k be Integer;
     assume A6: k = u+1;
     hence (-k) '*' a= ((0-u)-1)'*'a
                    .= (0-u) '*' a - a by Lm6
                    .= -(u '*' a) - a by A4
                    .= -(u '*' a + a) by RLVECT_1:30
                    .= -(k '*' a) by Lm5,A6;
     end;
   hence P[u+1];
   end;
 for i being Integer holds P[i] from INT_1:sch 4(A2,A3);
hence thesis;
end;
