let i1, i2, i3, i5 be Integer; :: thesis: ( i1,i2 are_congruent_mod i5 & i1,i3 are_congruent_mod i5 implies i2,i3 are_congruent_mod i5 )
assume that
A1: i1,i2 are_congruent_mod i5 and
A2: i1,i3 are_congruent_mod i5 ; :: thesis: i2,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 i2,i3 are_congruent_mod i5 ; :: thesis: verum