set ab = subrelstr [#a,b#];
let x, y be Element of L; :: according to YELLOW_0:def 16 :: thesis: ( not x in the carrier of (subrelstr [#a,b#]) or not y in the carrier of (subrelstr [#a,b#]) or not ex_inf_of {x,y},L or "/\" {x,y},L in the carrier of (subrelstr [#a,b#]) )
assume ( x in the carrier of (subrelstr [#a,b#]) & y in the carrier of (subrelstr [#a,b#]) & ex_inf_of {x,y},L ) ; :: thesis: "/\" {x,y},L in the carrier of (subrelstr [#a,b#])
then ( x in [#a,b#] & y in [#a,b#] ) by YELLOW_0:def 15;
then A1: ( x <= b & a <= x & a <= y ) by Def4;
A2: inf {x,y} = x "/\" y by YELLOW_0:40;
then ( inf {x,y} <= x & inf {x,y} <= y ) by YELLOW_0:23;
then A3: inf {x,y} <= b by A1, YELLOW_0:def 2;
a <= inf {x,y} by A1, A2, YELLOW_0:23;
then inf {x,y} in [#a,b#] by A3, Def4;
hence "/\" {x,y},L in the carrier of (subrelstr [#a,b#]) by YELLOW_0:def 15; :: thesis: verum