let I be set ; :: thesis: for x, X being ManySortedSet of I st not I is empty holds
{x} \/ X <> [[0]] I
let x, X be ManySortedSet of I; :: thesis: ( not I is empty implies {x} \/ X <> [[0]] I )
assume that
A1: not I is empty and
A2: {x} \/ X = [[0]] I ; :: thesis: contradiction
consider i being set such that
A3: i in I by A1, XBOOLE_0:def 1;
{(x . i)} \/ (X . i) <> {} ;
then ({x} . i) \/ (X . i) <> {} by A3, Def1;
then ({x} \/ X) . i <> {} by A3, PBOOLE:def 4;
hence contradiction by A2, PBOOLE:5; :: thesis: verum