let L1, L2 be RelStr ; :: thesis: ( RelStr(# the carrier of L1,the InternalRel of L1 #) = RelStr(# the carrier of L2,the InternalRel of L2 #) implies for X1 being Subset of L1
for X2 being Subset of L2 st X1 = X2 holds
( ( X1 is lower implies X2 is lower ) & ( X1 is upper implies X2 is upper ) ) )
assume A1: RelStr(# the carrier of L1,the InternalRel of L1 #) = RelStr(# the carrier of L2,the InternalRel of L2 #) ; :: thesis: for X1 being Subset of L1
for X2 being Subset of L2 st X1 = X2 holds
( ( X1 is lower implies X2 is lower ) & ( X1 is upper implies X2 is upper ) )
let X1 be Subset of L1; :: thesis: for X2 being Subset of L2 st X1 = X2 holds
( ( X1 is lower implies X2 is lower ) & ( X1 is upper implies X2 is upper ) )
let X2 be Subset of L2; :: thesis: ( X1 = X2 implies ( ( X1 is lower implies X2 is lower ) & ( X1 is upper implies X2 is upper ) ) )
assume A2: X1 = X2 ; :: thesis: ( ( X1 is lower implies X2 is lower ) & ( X1 is upper implies X2 is upper ) )

hereby :: thesis: ( X1 is upper implies X2 is upper )

assume X1 is lower ; :: thesis: X2 is lower
then ( downarrow X1 = downarrow X2 & downarrow X1 c= X1 ) by A1, A2, Th13, Th23;
hence X2 is lower by A2, Th23; :: thesis: verum

end;

assume X1 is upper ; :: thesis: X2 is upper
then ( uparrow X1 = uparrow X2 & uparrow X1 c= X1 ) by A1, A2, Th13, Th24;
hence X2 is upper by A2, Th24; :: thesis: verum