let L2 be Lattice; :: thesis: for 1L being upper-bounded Lattice
for f being Homomorphism of 1L,L2 st f is onto holds
( L2 is upper-bounded & f preserves_top )
let 1L be upper-bounded Lattice; :: thesis: for f being Homomorphism of 1L,L2 st f is onto holds
( L2 is upper-bounded & f preserves_top )
let f be Homomorphism of 1L,L2; :: thesis: ( f is onto implies ( L2 is upper-bounded & f preserves_top ) )
set r = f . (Top 1L);
assume A1: f is onto ; :: thesis: ( L2 is upper-bounded & f preserves_top )

A2: now

let a2 be Element of L2; :: thesis: ( (f . (Top 1L)) "\/" a2 = f . (Top 1L) & a2 "\/" (f . (Top 1L)) = f . (Top 1L) )
consider a1 being Element of 1L such that
A3: f . a1 = a2 by A1, Th9;
thus (f . (Top 1L)) "\/" a2 = f . ((Top 1L) "\/" a1) by A3, Def1
.= f . (Top 1L) by LATTICES:18 ; :: thesis: a2 "\/" (f . (Top 1L)) = f . (Top 1L)
hence a2 "\/" (f . (Top 1L)) = f . (Top 1L) ; :: thesis: verum

end;

thus L2 is upper-bounded :: thesis: f preserves_top

proof

take f . (Top 1L) ; :: according to LATTICES:def 14 :: thesis: for b₁ being Element of the carrier of L2 holds
( (f . (Top 1L)) "\/" b₁ = f . (Top 1L) & b₁ "\/" (f . (Top 1L)) = f . (Top 1L) )
thus for b₁ being Element of the carrier of L2 holds
( (f . (Top 1L)) "\/" b₁ = f . (Top 1L) & b₁ "\/" (f . (Top 1L)) = f . (Top 1L) ) by A2; :: thesis: verum

end;

then Top L2 = f . (Top 1L) by A2, LATTICES:def 17;
hence f preserves_top by Def7; :: thesis: verum