defpred S₁[ set ] means ex a being non negative real number ex R being Equivalence_Relation of M st
( R = (dist_toler M,a) [*] & Class R = $1 );
let X1, X2 be Subset of (PARTITIONS the carrier of M); :: thesis: ( ( for x being set holds
( x in X1 iff ex a being non negative real number ex R being Equivalence_Relation of M st
( R = (dist_toler M,a) [*] & Class R = x ) ) ) & ( for x being set holds
( x in X2 iff ex a being non negative real number ex R being Equivalence_Relation of M st
( R = (dist_toler M,a) [*] & Class R = x ) ) ) implies X1 = X2 )
assume A3: ( ( for x being set holds
( x in X1 iff ex a being non negative real number ex R being Equivalence_Relation of M st
( R = (dist_toler M,a) [*] & Class R = x ) ) ) & ( for x being set holds
( x in X2 iff ex a being non negative real number ex R being Equivalence_Relation of M st
( R = (dist_toler M,a) [*] & Class R = x ) ) ) ) ; :: thesis: X1 = X2
for X1, X2 being Subset of (PARTITIONS the carrier of M) st ( for x being set holds
( x in X1 iff S₁[x] ) ) & ( for x being set holds
( x in X2 iff S₁[x] ) ) holds
X1 = X2 from LMOD_7:sch 1();
hence X1 = X2 by A3; :: thesis: verum