let T1, T2 be non empty RelStr ; :: thesis: for S1 being non empty full SubRelStr of T1
for S2 being non empty full SubRelStr of T2 st RelStr(# the carrier of T1, the InternalRel of T1 #) = RelStr(# the carrier of T2, the InternalRel of T2 #) & the carrier of S1 = the carrier of S2 & S1 is sups-inheriting holds
S2 is sups-inheriting
let S1 be non empty full SubRelStr of T1; :: thesis: for S2 being non empty full SubRelStr of T2 st RelStr(# the carrier of T1, the InternalRel of T1 #) = RelStr(# the carrier of T2, the InternalRel of T2 #) & the carrier of S1 = the carrier of S2 & S1 is sups-inheriting holds
S2 is sups-inheriting
let S2 be non empty full SubRelStr of T2; :: thesis: ( RelStr(# the carrier of T1, the InternalRel of T1 #) = RelStr(# the carrier of T2, the InternalRel of T2 #) & the carrier of S1 = the carrier of S2 & S1 is sups-inheriting implies S2 is sups-inheriting )
assume A1: RelStr(# the carrier of T1, the InternalRel of T1 #) = RelStr(# the carrier of T2, the InternalRel of T2 #) ; :: thesis: ( not the carrier of S1 = the carrier of S2 or not S1 is sups-inheriting or S2 is sups-inheriting )
assume A2: the carrier of S1 = the carrier of S2 ; :: thesis: ( not S1 is sups-inheriting or S2 is sups-inheriting )
assume A3: for X being Subset of S1 st ex_sup_of X,T1 holds
"\/" (X,T1) in the carrier of S1 ; :: according to YELLOW_0:def 19 :: thesis: S2 is sups-inheriting
let X be Subset of S2; :: according to YELLOW_0:def 19 :: thesis: ( not ex_sup_of X,T2 or "\/" (X,T2) in the carrier of S2 )
reconsider Y = X as Subset of S1 by A2;
assume A4: ex_sup_of X,T2 ; :: thesis: "\/" (X,T2) in the carrier of S2
then "\/" (Y,T1) in the carrier of S1 by A1, A3, YELLOW_0:14;
hence "\/" (X,T2) in the carrier of S2 by A1, A2, A4, YELLOW_0:26; :: thesis: verum