let A be non empty set ; :: thesis: for B being Finite_Subset of A
for L being 0_Lattice
for f, g being Function of A,the carrier of L st ( for x being Element of A st x in B holds
f . x [= g . x ) holds
FinJoin B,f [= FinJoin B,g

let B be Finite_Subset of A; :: thesis: for L being 0_Lattice
for f, g being Function of A,the carrier of L st ( for x being Element of A st x in B holds
f . x [= g . x ) holds
FinJoin B,f [= FinJoin B,g

let L be 0_Lattice; :: thesis: for f, g being Function of A,the carrier of L st ( for x being Element of A st x in B holds
f . x [= g . x ) holds
FinJoin B,f [= FinJoin B,g

let f, g be Function of A,the carrier of L; :: thesis: ( ( for x being Element of A st x in B holds
f . x [= g . x ) implies FinJoin B,f [= FinJoin B,g )

assume A1: for x being Element of A st x in B holds
f . x [= g . x ; :: thesis: FinJoin B,f [= FinJoin B,g
now
let x be Element of A; :: thesis: ( x in B implies f . x [= FinJoin B,g )
assume A2: x in B ; :: thesis: f . x [= FinJoin B,g
then f . x [= g . x by A1;
hence f . x [= FinJoin B,g by A2, Th44; :: thesis: verum
end;
hence FinJoin B,f [= FinJoin B,g by Th70; :: thesis: verum