reconsider MM = MetrStruct(# the carrier of M,the distance of M #) as MetrSpace by Lm1;
for x, y being Point of MM holds the distance of MM . x,y = the distance of M . x,y ;
then MM is SubSpace of M by Def1;
hence ex b₁ being SubSpace of M st
( b₁ is strict & not b₁ is empty ) ; :: thesis: verum