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 Th83:
for R being Ring, f being Homomorphism of INT.Ring,R holds f = canHom_Int R
proof
let R be Ring, f being Homomorphism of INT.Ring,R;
set g = canHom_Int R;
A1: dom f = the carrier of I by FUNCT_2:def 1
        .= dom g by FUNCT_2:def 1;
defpred P[Integer] means
  for j be Integer st j = $1 holds f.j = j '*' 1.R;
now let j be Integer;
  assume A2: j = 0;
  hence f.j = f.(0.INT.Ring)
           .= 0.R by RING_2:6
           .= j '*' 1.R by A2,Th58;
  end;
then A3: P[0];
A4: for u being Integer holds P[u] implies P[u - 1] & P[u + 1]
   proof
   let u be Integer;
   assume A5: P[u];
   reconsider uu = u as Element of INT.Ring by INT_1:def 2;
   now let k be Integer;
     assume A6: k = u-1;
     then k = uu - 1.I;
     hence f.k = f.uu - f.(1.I) by RING_2:8
              .= (uu '*' 1.R) - f.(1_I) by A5
              .= (uu '*' 1.R) - 1_R by GROUP_1:def 13
              .= (uu '*' 1.R) - (1 '*' 1.R) by Th59
              .= k '*' 1.R by Th63,A6;
     end;
   hence P[u-1];
   now let k be Integer;
     assume A7: k = u+1;
     then k = uu + 1.I;
     hence f.k = f.uu + f.(1.I) by VECTSP_1:def 20
              .= (uu '*' 1.R) + f.(1_I) by A5
              .= (uu '*' 1.R) + 1_R by GROUP_1:def 13
              .= (uu '*' 1.R) + (1 '*' 1.R) by Th59
              .= k '*' 1.R by Th61,A7;
     end;
   hence P[u+1];
   end;
A8: for i be Integer holds P[i] from INT_1:sch 4(A3,A4);
now let x be object;
  assume x in dom f;
  then reconsider a = x as Element of INT.Ring;
  reconsider aa = a as Integer;
  f.a = aa '*' 1.R by A8;
  hence f.x = g.x by Def8;
  end;
hence thesis by A1;
end;
