reserve Y for RealNormSpace;
reserve X,Y for RealBanachSpace;
reserve Z for open Subset of REAL;
reserve a,b,c,d,e,r,x0 for Real;
reserve y0 for VECTOR of X;
reserve G for Function of X,X;

theorem Th45a:
for f be PartFunc of REAL,the carrier of X
  st a <= b & [.a,b.] c= dom f
   & (for x be Real st x in [.a,b.] holds f is_continuous_in x)
   & f is_differentiable_on ].a,b.[
   & for x be Real st x in ].a,b.[ holds diff(f,x) = 0.X holds
  f/.b = f/.a
proof
   let f be PartFunc of REAL,the carrier of X;
   assume that
A1: a <= b & [.a,b.] c= dom f
  & for x be Real st x in [.a,b.] holds f is_continuous_in x and
A2: f is_differentiable_on ].a,b.[ and
A3: for x be Real st x in ].a,b.[ holds diff(f,x) = 0.X;
A5:for x be Real st x in ].a,b.[ holds f is_differentiable_in x
     by A2,NDIFF_3:10;
   now let x be Real;
    assume x in ].a,b.[; then
    ||.diff(f,x).|| = ||. 0.X .|| by A3;
    hence ||.diff(f,x).|| <= 0;
   end; then
   ||. f/.b - f/.a .|| <= 0*|.b-a.| by A5,Th519,A1; then
   ||. f/.b - f/.a .|| = 0;
   hence f/.b = f/.a by NORMSP_1:6;
end;
