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
for R being Ring, S being R-homomorphic Ring, f being Homomorphism of R,S
for a being Element of R,
    i being Integer holds f.(i '*' a) = i '*' f.a
proof
let R be Ring,
    S be R-homomorphic Ring,
    f be Homomorphism of R,S;
let a be Element of R, i be Integer;
defpred P[Integer] means
  for j being Integer st j = $1 holds f.(j'*'a) = j'*'f.a;
now let j be Integer;
  assume A1: j = 0;
  hence f.(j'*'a) = f.(0.R) by Th58
                 .= 0.S by RING_2:6
                 .= j '*' f.a by A1,Th58;
  end;
then A2: P[0];
A3: for i being Integer holds P[i] implies P[i - 1] & P[i + 1]
   proof
   let i be Integer;
   assume A4: P[i];
   now let k be Integer;
     assume k = i-1;
     then A5: f.((k+1)'*'a) = (k+1) '*' f.a by A4
                           .= (k '*' (f.a)) + (1 '*' f.a) by Th61
                           .= (k '*' (f.a)) + f.a by Th59;
     f.((k+1)'*'a) = f.(k'*'a + 1'*'a) by Th61
                  .= f.(k'*'a + a) by Th59
                  .= f.(k'*'a) + f.a by VECTSP_1:def 20;
     hence f.(k '*' a) = k '*' f.a by A5,RLVECT_1:8;
     end;
   hence P[i-1];
   now let k be Integer;
     assume k = i+1;
     then A6: f.((k-1) '*' a) = (k-1) '*' f.a by A4
                   .= k '*' (f.a) + (-1.(INT.Ring))'*' f.a by Th61
                   .= k '*' (f.a) - f.a by Th60;
     f.((k-1)'*'a) = f.(k'*'a + (-1.(INT.Ring))'*'a) by Th61
                  .= f.(k'*'a - a) by Th60
                  .= f.(k'*'a) - f.a by RING_2:8;
     hence f.(k '*' a) = k '*' f.a by A6,RLVECT_1:8;
     end;
   hence P[i+1];
   end;
for i being Integer holds P[i] from INT_1:sch 4(A2,A3);
hence thesis;
end;
