let i1, i2, i3, i4, i5 be Integer; :: thesis: ( i1,i2 are_congruent_mod i5 & i3,i4 are_congruent_mod i5 implies i1 - i3,i2 - i4 are_congruent_mod i5 )
assume that
A1: i1,i2 are_congruent_mod i5 and
A2: i3,i4 are_congruent_mod i5 ; :: thesis: i1 - i3,i2 - i4 are_congruent_mod i5
consider i8 being Integer such that
A3: i5 * i8 = i1 - i2 by A1;
consider i9 being Integer such that
A4: i5 * i9 = i3 - i4 by A2;
(i1 - i3) - (i2 - i4) = i5 * (i8 - i9) by A3, A4;
hence i1 - i3,i2 - i4 are_congruent_mod i5 ; :: thesis: verum