let X1, X2 be non empty set ; :: thesis: for cB1 being filter_base of X1
for cB2 being filter_base of X2
for cF1 being Filter of X1
for cF2 being Filter of X2
for Y being non empty TopSpace
for x being Point of Y
for f being Function of [:X1,X2:],Y st x in lim_filter (f,<.cF1,cF2.)) & <.cB1.) = cF1 & <.cB2.) = cF2 holds
for U being a_neighborhood of x st U is closed holds
ex B1 being Element of cB1 ex B2 being Element of cB2 st
for y being Element of B1 holds f .: [:{y},B2:] c= Int U
let cB1 be filter_base of X1; :: thesis: for cB2 being filter_base of X2
for cF1 being Filter of X1
for cF2 being Filter of X2
for Y being non empty TopSpace
for x being Point of Y
for f being Function of [:X1,X2:],Y st x in lim_filter (f,<.cF1,cF2.)) & <.cB1.) = cF1 & <.cB2.) = cF2 holds
for U being a_neighborhood of x st U is closed holds
ex B1 being Element of cB1 ex B2 being Element of cB2 st
for y being Element of B1 holds f .: [:{y},B2:] c= Int U
let cB2 be filter_base of X2; :: thesis: for cF1 being Filter of X1
for cF2 being Filter of X2
for Y being non empty TopSpace
for x being Point of Y
for f being Function of [:X1,X2:],Y st x in lim_filter (f,<.cF1,cF2.)) & <.cB1.) = cF1 & <.cB2.) = cF2 holds
for U being a_neighborhood of x st U is closed holds
ex B1 being Element of cB1 ex B2 being Element of cB2 st
for y being Element of B1 holds f .: [:{y},B2:] c= Int U
let cF1 be Filter of X1; :: thesis: for cF2 being Filter of X2
for Y being non empty TopSpace
for x being Point of Y
for f being Function of [:X1,X2:],Y st x in lim_filter (f,<.cF1,cF2.)) & <.cB1.) = cF1 & <.cB2.) = cF2 holds
for U being a_neighborhood of x st U is closed holds
ex B1 being Element of cB1 ex B2 being Element of cB2 st
for y being Element of B1 holds f .: [:{y},B2:] c= Int U
let cF2 be Filter of X2; :: thesis: for Y being non empty TopSpace
for x being Point of Y
for f being Function of [:X1,X2:],Y st x in lim_filter (f,<.cF1,cF2.)) & <.cB1.) = cF1 & <.cB2.) = cF2 holds
for U being a_neighborhood of x st U is closed holds
ex B1 being Element of cB1 ex B2 being Element of cB2 st
for y being Element of B1 holds f .: [:{y},B2:] c= Int U
let Y be non empty TopSpace; :: thesis: for x being Point of Y
for f being Function of [:X1,X2:],Y st x in lim_filter (f,<.cF1,cF2.)) & <.cB1.) = cF1 & <.cB2.) = cF2 holds
for U being a_neighborhood of x st U is closed holds
ex B1 being Element of cB1 ex B2 being Element of cB2 st
for y being Element of B1 holds f .: [:{y},B2:] c= Int U
let x be Point of Y; :: thesis: for f being Function of [:X1,X2:],Y st x in lim_filter (f,<.cF1,cF2.)) & <.cB1.) = cF1 & <.cB2.) = cF2 holds
for U being a_neighborhood of x st U is closed holds
ex B1 being Element of cB1 ex B2 being Element of cB2 st
for y being Element of B1 holds f .: [:{y},B2:] c= Int U
let f be Function of [:X1,X2:],Y; :: thesis: ( x in lim_filter (f,<.cF1,cF2.)) & <.cB1.) = cF1 & <.cB2.) = cF2 implies for U being a_neighborhood of x st U is closed holds
ex B1 being Element of cB1 ex B2 being Element of cB2 st
for y being Element of B1 holds f .: [:{y},B2:] c= Int U )
assume that
A1: x in lim_filter (f,<.cF1,cF2.)) and
A2: <.cB1.) = cF1 and
A3: <.cB2.) = cF2 ; :: thesis: for U being a_neighborhood of x st U is closed holds
ex B1 being Element of cB1 ex B2 being Element of cB2 st
for y being Element of B1 holds f .: [:{y},B2:] c= Int U
now :: thesis: for U being a_neighborhood of x st U is closed holds
ex B1 being Element of cB1 ex B2 being Element of cB2 st
for y being Element of B1 holds f .: [:{y},B2:] c= Int U
let U be a_neighborhood of x; :: thesis: ( U is closed implies ex B1 being Element of cB1 ex B2 being Element of cB2 st
for y being Element of B1 holds f .: [:{y},B2:] c= Int U )
assume U is closed ; :: thesis: ex B1 being Element of cB1 ex B2 being Element of cB2 st
for y being Element of B1 holds f .: [:{y},B2:] c= Int U
then consider B1 being Element of cB1, B2 being Element of cB2 such that
A4: f .: [:B1,B2:] c= Int U by A1, A2, A3, Th75;
take B1 = B1; :: thesis: ex B2 being Element of cB2 st
for y being Element of B1 holds f .: [:{y},B2:] c= Int U
take B2 = B2; :: thesis: for y being Element of B1 holds f .: [:{y},B2:] c= Int U
let y be Element of B1; :: thesis: f .: [:{y},B2:] c= Int U
[:{y},B2:] c= [:B1,B2:] by ZFMISC_1:95;
then f .: [:{y},B2:] c= f .: [:B1,B2:] by RELAT_1:125;
hence f .: [:{y},B2:] c= Int U by A4; :: thesis: verum
end;
hence for U being a_neighborhood of x st U is closed holds
ex B1 being Element of cB1 ex B2 being Element of cB2 st
for y being Element of B1 holds f .: [:{y},B2:] c= Int U ; :: thesis: verum