let C be complete Lattice; :: thesis: for a being Element of C
for X being set st a in X & X is_less_than a holds
"\/" (X,C) = a
let a be Element of C; :: thesis: for X being set st a in X & X is_less_than a holds
"\/" (X,C) = a
let X be set ; :: thesis: ( a in X & X is_less_than a implies "\/" (X,C) = a )
assume that
A1: a in X and
A2: X is_less_than a ; :: thesis: "\/" (X,C) = a
A3: "\/" (X,C) [= a by A2, Def21;
a [= "\/" (X,C) by A1, Th38;
hence "\/" (X,C) = a by A3, LATTICES:8; :: thesis: verum