let z0 be Complex; :: thesis: for N1, N2 being Neighbourhood of z0 ex N being Neighbourhood of z0 st
( N c= N1 & N c= N2 )
let N1, N2 be Neighbourhood of z0; :: thesis: ex N being Neighbourhood of z0 st
( N c= N1 & N c= N2 )
consider a1 being Real such that
A1: 0 < a1 and
A2: { y where y is Complex : |.(y - z0).| < a1 } c= N1 by Def5;
consider a2 being Real such that
A3: 0 < a2 and
A4: { y where y is Complex : |.(y - z0).| < a2 } c= N2 by Def5;
set g = min (a1,a2);
take IT = { y where y is Complex : |.(y - z0).| < min (a1,a2) } ; :: thesis: ( IT is Subset of COMPLEX & IT is Neighbourhood of z0 & IT c= N1 & IT c= N2 )
A5: min (a1,a2) <= a2 by XXREAL_0:17;
A6: { y where y is Complex : |.(y - z0).| < min (a1,a2) } c= { y where y is Complex : |.(y - z0).| < a2 }

proof

let z be object ; :: according to TARSKI:def 3 :: thesis: ( not z in { y where y is Complex : |.(y - z0).| < min (a1,a2) } or z in { y where y is Complex : |.(y - z0).| < a2 } )
assume z in { y where y is Complex : |.(y - z0).| < min (a1,a2) } ; :: thesis: z in { y where y is Complex : |.(y - z0).| < a2 }
then consider y being Complex such that
A7: z = y and
A8: |.(y - z0).| < min (a1,a2) ;
|.(y - z0).| < a2 by A5, A8, XXREAL_0:2;
hence z in { y where y is Complex : |.(y - z0).| < a2 } by A7; :: thesis: verum

end;

A9: min (a1,a2) <= a1 by XXREAL_0:17;
A10: { y where y is Complex : |.(y - z0).| < min (a1,a2) } c= { y where y is Complex : |.(y - z0).| < a1 }

proof

let z be object ; :: according to TARSKI:def 3 :: thesis: ( not z in { y where y is Complex : |.(y - z0).| < min (a1,a2) } or z in { y where y is Complex : |.(y - z0).| < a1 } )
assume z in { y where y is Complex : |.(y - z0).| < min (a1,a2) } ; :: thesis: z in { y where y is Complex : |.(y - z0).| < a1 }
then consider y being Complex such that
A11: z = y and
A12: |.(y - z0).| < min (a1,a2) ;
|.(y - z0).| < a1 by A9, A12, XXREAL_0:2;
hence z in { y where y is Complex : |.(y - z0).| < a1 } by A11; :: thesis: verum

end;

0 < min (a1,a2) by A1, A3, XXREAL_0:15;
hence ( IT is Subset of COMPLEX & IT is Neighbourhood of z0 & IT c= N1 & IT c= N2 ) by A2, A4, A10, A6, Th6; :: thesis: verum