reserve d,i,j,k,m,n,p,q,x,k1,k2 for Nat,
  a,c,i1,i2,i3,i5 for Integer;

theorem Th40:
  i1,i2 are_congruent_mod i5 & i1,i3 are_congruent_mod i5 implies
  i2,i3 are_congruent_mod i5
proof
  assume that
A1: i1,i2 are_congruent_mod i5 and
A2: i1,i3 are_congruent_mod i5;
  i2,i1 are_congruent_mod i5 by A1,INT_1:14;
  then consider t1 being Integer such that
A3: i5*t1 = i2 - i1;
  i3,i1 are_congruent_mod i5 by A2,INT_1:14;
  then consider t2 being Integer such that
A4: i5*t2 = i3 - i1;
  i2 - i3 = i5*(t1 - t2) by A3,A4;
  hence thesis;
end;
