let O be set ; :: thesis: for G being GroupWithOperators of O
for H1 being StableSubgroup of G
for H2 being strict StableSubgroup of G holds
( H1 is StableSubgroup of H2 iff H1 "\/" H2 = H2 )
let G be GroupWithOperators of O; :: thesis: for H1 being StableSubgroup of G
for H2 being strict StableSubgroup of G holds
( H1 is StableSubgroup of H2 iff H1 "\/" H2 = H2 )
let H1 be StableSubgroup of G; :: thesis: for H2 being strict StableSubgroup of G holds
( H1 is StableSubgroup of H2 iff H1 "\/" H2 = H2 )
let H2 be strict StableSubgroup of G; :: thesis: ( H1 is StableSubgroup of H2 iff H1 "\/" H2 = H2 )
thus ( H1 is StableSubgroup of H2 implies H1 "\/" H2 = H2 ) :: thesis: ( H1 "\/" H2 = H2 implies H1 is StableSubgroup of H2 )
proof
assume H1 is StableSubgroup of H2 ; :: thesis: H1 "\/" H2 = H2
then H1 is Subgroup of H2 by Def7;
then the carrier of H1 c= the carrier of H2 by GROUP_2:def 5;
hence H1 "\/" H2 = the_stable_subgroup_of (carr H2) by XBOOLE_1:12
.= H2 by Th25 ;
:: thesis: verum
end;
thus ( H1 "\/" H2 = H2 implies H1 is StableSubgroup of H2 ) by Th35; :: thesis: verum