let X, Y be ComplexNormSpace; :: thesis: for f being Point of
for g being Lipschitzian LinearOperator of X,Y st g = f holds
for t being VECTOR of X holds ||.(g . t).|| <= *

let f be Point of ; :: thesis: for g being Lipschitzian LinearOperator of X,Y st g = f holds
for t being VECTOR of X holds ||.(g . t).|| <= *

let g be Lipschitzian LinearOperator of X,Y; :: thesis: ( g = f implies for t being VECTOR of X holds ||.(g . t).|| <= * )
assume A1: g = f ; :: thesis: for t being VECTOR of X holds ||.(g . t).|| <= *
A2: ( not PreNorms g is empty & PreNorms g is bounded_above ) by Th26;
now :: thesis: for t being VECTOR of X holds ||.(g . t).|| <= *
let t be VECTOR of X; :: thesis: ||.(g . t).|| <= *
now :: thesis: ( ( t = 0. X & ||.(g . t).|| <= * ) or ( t <> 0. X & ||.(g . t).|| <= * ) )
per cases ( t = 0. X or t <> 0. X ) ;
case A3: t = 0. X ; :: thesis: ||.(g . t).|| <= *
then A4: ||.t.|| = 0 by NORMSP_0:def 6;
g . t = g . (0c * (0. X)) by
.= 0c * (g . (0. X)) by Def3
.= 0. Y by CLVECT_1:1 ;
hence ||.(g . t).|| <= * by ; :: thesis: verum
end;
case A5: t <> 0. X ; :: thesis: ||.(g . t).|| <= *
reconsider t0 = () + () as Element of COMPLEX by XCMPLX_0:def 2;
reconsider t1 = t0 * t as VECTOR of X ;
A6: ||.t.|| <> 0 by ;
then A7: ||.t.|| > 0 by CLVECT_1:105;
A8: |.t0.| = |.(1 * ()).|
.= |.(1 / ).| by XCMPLX_0:def 9
.= 1 / by ABSVALUE:7
.= 1 / by
.= 1 * () by XCMPLX_0:def 9
.= " ;
then A9: ||.(g . t).|| / = ||.(g . t).|| * |.t0.| by XCMPLX_0:def 9
.= ||.(t0 * (g . t)).|| by CLVECT_1:def 13
.= ||.(g . t1).|| by Def3 ;
||.t1.|| = |.t0.| * by CLVECT_1:def 13
.= 1 by ;
then ||.(g . t).|| / in { ||.(g . s).|| where s is VECTOR of X : <= 1 } by A9;
then ||.(g . t).|| / <= upper_bound () by ;
then ||.(g . t).|| / <= () . g by Th29;
then A10: ||.(g . t).|| / <= by A1;
(||.(g . t).|| / ) * = (||.(g . t).|| * ()) * by XCMPLX_0:def 9
.= ||.(g . t).|| * (() * )
.= ||.(g . t).|| * 1 by
.= ||.(g . t).|| ;
hence ||.(g . t).|| <= * by ; :: thesis: verum
end;
end;
end;
hence ||.(g . t).|| <= * ; :: thesis: verum
end;
hence for t being VECTOR of X holds ||.(g . t).|| <= * ; :: thesis: verum