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 & X1 is directed holds
X2 is directed )
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 & X1 is directed holds
X2 is directed
let X1 be Subset of L1; :: thesis: for X2 being Subset of L2 st X1 = X2 & X1 is directed holds
X2 is directed
let X2 be Subset of L2; :: thesis: ( X1 = X2 & X1 is directed implies X2 is directed )
assume A2: X1 = X2 ; :: thesis: ( not X1 is directed or X2 is directed )
assume A3: for x, y being Element of L1 st x in X1 & y in X1 holds
ex z being Element of L1 st
( z in X1 & x <= z & y <= z ) ; :: according to WAYBEL_0:def 1 :: thesis: X2 is directed
let x, y be Element of L2; :: according to WAYBEL_0:def 1 :: thesis: ( x in X2 & y in X2 implies ex z being Element of L2 st
( z in X2 & x <= z & y <= z ) )
reconsider x' = x, y' = y as Element of L1 by A1;
assume ( x in X2 & y in X2 ) ; :: thesis: ex z being Element of L2 st
( z in X2 & x <= z & y <= z )
then consider z' being Element of L1 such that
A4: ( z' in X1 & x' <= z' & y' <= z' ) by A2, A3;
reconsider z = z' as Element of L2 by A1;
take z ; :: thesis: ( z in X2 & x <= z & y <= z )
thus ( z in X2 & x <= z & y <= z ) by A1, A2, A4, YELLOW_0:1; :: thesis: verum