{A} is a_partition of A
by EQREL_1:39;

then {A} in PARTITIONS A by PARTIT1:def 3;

then reconsider S = {{A}} as Subset of (PARTITIONS A) by ZFMISC_1:31;

take S ; :: thesis: for X, Y being a_partition of A st X in S & Y in S & not X is_finer_than Y holds

Y is_finer_than X

thus for X, Y being a_partition of A st X in S & Y in S & not X is_finer_than Y holds

Y is_finer_than X by Lm1; :: thesis: verum

then {A} in PARTITIONS A by PARTIT1:def 3;

then reconsider S = {{A}} as Subset of (PARTITIONS A) by ZFMISC_1:31;

take S ; :: thesis: for X, Y being a_partition of A st X in S & Y in S & not X is_finer_than Y holds

Y is_finer_than X

thus for X, Y being a_partition of A st X in S & Y in S & not X is_finer_than Y holds

Y is_finer_than X by Lm1; :: thesis: verum