reserve x for object;
reserve x0,r,r1,r2,g,g1,g2,p,y0 for Real;
reserve n,m,k,l for Element of NAT;
reserve a,b,d for Real_Sequence;
reserve h,h1,h2 for non-zero 0-convergent Real_Sequence;
reserve c,c1 for constant Real_Sequence;
reserve A for open Subset of REAL;
reserve f,f1,f2 for PartFunc of REAL,REAL;
reserve L for LinearFunc;
reserve R for RestFunc;

theorem Th25:
  (for r1,r2 st r1 in ].p,g.[ & r2 in ].p,g.[ holds
    |.f.r1 - f.r2.| <= (r1 - r2)^2) & ].p,g.[ c= dom f
  implies f is_differentiable_on ].p,g.[ & f|].p,g.[ is constant
proof
  assume that
A1: for r1,r2 st r1 in ].p,g.[ & r2 in ].p,g.[ holds |.f.r1-f.r2.|<=(r1
  -r2)^2 and
A2: ].p,g.[ c= dom f;
  thus
A3: f is_differentiable_on ].p,g.[ by A1,A2,Th24;
  for x0 st x0 in ].p,g.[ holds diff(f,x0)=0 by A1,A2,Th24;
  hence thesis by A3,ROLLE:7;
end;
