let p, g be Real; :: thesis: for f being PartFunc of REAL,REAL st ].p,g.[ c= dom f & f is_differentiable_on ].p,g.[ & ( for x being Real st x in ].p,g.[ holds
diff (f,x) = 0 ) holds
f | ].p,g.[ is V8()
let f be PartFunc of REAL,REAL; :: thesis: ( ].p,g.[ c= dom f & f is_differentiable_on ].p,g.[ & ( for x being Real st x in ].p,g.[ holds
diff (f,x) = 0 ) implies f | ].p,g.[ is V8() )
assume that
A1: ].p,g.[ c= dom f and
A2: f is_differentiable_on ].p,g.[ and
A3: for x being Real st x in ].p,g.[ holds
diff (f,x) = 0 ; :: thesis: f | ].p,g.[ is V8()
now :: thesis: for x1, x2 being Element of REAL st x1 in ].p,g.[ /\ (dom f) & x2 in ].p,g.[ /\ (dom f) holds
f . x1 = f . x2
let x1, x2 be Element of REAL ; :: thesis: ( x1 in ].p,g.[ /\ (dom f) & x2 in ].p,g.[ /\ (dom f) implies f . x1 = f . x2 )
assume ( x1 in ].p,g.[ /\ (dom f) & x2 in ].p,g.[ /\ (dom f) ) ; :: thesis: f . x1 = f . x2
then A4: ( x1 in ].p,g.[ & x2 in ].p,g.[ ) by XBOOLE_0:def 4;
now :: thesis: f . x1 = f . x2
per cases ( x1 = x2 or not x1 = x2 ) ;
suppose x1 = x2 ; :: thesis: f . x1 = f . x2
hence f . x1 = f . x2 ; :: thesis: verum
end;
suppose A5: not x1 = x2 ; :: thesis: f . x1 = f . x2
now :: thesis: f . x1 = f . x2
per cases ( x1 < x2 or x2 < x1 ) by A5, XXREAL_0:1;
suppose A6: x1 < x2 ; :: thesis: f . x1 = f . x2
then 0 <> x2 - x1 ;
then A7: 0 <> (x2 - x1) " by XCMPLX_1:202;
reconsider Z = ].x1,x2.[ as open Subset of REAL ;
A8: [.x1,x2.] c= ].p,g.[ by A4, XXREAL_2:def 12;
f | ].p,g.[ is continuous by A2, FDIFF_1:25;
then A9: f | [.x1,x2.] is continuous by A8, FCONT_1:16;
A10: Z c= [.x1,x2.] by XXREAL_1:25;
then f is_differentiable_on Z by A2, A8, FDIFF_1:26, XBOOLE_1:1;
then A11: ex x0 being Real st
( x0 in ].x1,x2.[ & diff (f,x0) = ((f . x2) - (f . x1)) / (x2 - x1) ) by A1, A6, A8, A9, Th3, XBOOLE_1:1;
Z c= ].p,g.[ by A8, A10;
then 0 = (f . x2) - (f . x1) by A3, A7, A11, XCMPLX_1:6;
hence f . x1 = f . x2 ; :: thesis: verum
end;
suppose A12: x2 < x1 ; :: thesis: f . x1 = f . x2
then 0 <> x1 - x2 ;
then A13: 0 <> (x1 - x2) " by XCMPLX_1:202;
reconsider Z = ].x2,x1.[ as open Subset of REAL ;
A14: [.x2,x1.] c= ].p,g.[ by A4, XXREAL_2:def 12;
f | ].p,g.[ is continuous by A2, FDIFF_1:25;
then A15: f | [.x2,x1.] is continuous by A14, FCONT_1:16;
A16: Z c= [.x2,x1.] by XXREAL_1:25;
then f is_differentiable_on Z by A2, A14, FDIFF_1:26, XBOOLE_1:1;
then A17: ex x0 being Real st
( x0 in ].x2,x1.[ & diff (f,x0) = ((f . x1) - (f . x2)) / (x1 - x2) ) by A1, A12, A14, A15, Th3, XBOOLE_1:1;
Z c= ].p,g.[ by A14, A16;
then 0 = (f . x1) - (f . x2) by A3, A13, A17, XCMPLX_1:6;
hence f . x1 = f . x2 ; :: thesis: verum
end;
end;
end;
hence f . x1 = f . x2 ; :: thesis: verum
end;
end;
end;
hence f . x1 = f . x2 ; :: thesis: verum
end;
hence f | ].p,g.[ is V8() by PARTFUN2:58; :: thesis: verum