let r be Real; :: thesis: ( ].0 ,r.[ c= dom ((#Z 2) ^ ) implies ((#Z 1) ^ ) `| ].0 ,r.[ = ((- 1) (#) ((#Z 2) ^ )) | ].0 ,r.[ )
assume A2: ].0 ,r.[ c= dom ((#Z 2) ^ ) ; :: thesis: ((#Z 1) ^ ) `| ].0 ,r.[ = ((- 1) (#) ((#Z 2) ^ )) | ].0 ,r.[
(#Z 1) ^ is_differentiable_on ].0 ,r.[ by Th34;
then A4: dom (((#Z 1) ^ ) `| ].0 ,r.[) = ].0 ,r.[ by FDIFF_1:def 8;
A5: dom (((- 1) (#) ((#Z (1 + 1)) ^ )) | ].0 ,r.[) = (dom ((- 1) (#) ((#Z 2) ^ ))) /\ ].0 ,r.[ by FUNCT_1:68
.= (dom ((#Z 2) ^ )) /\ ].0 ,r.[ by VALUED_1:def 5
.= ].0 ,r.[ by A2, XBOOLE_1:28 ;
for x0 being Real st x0 in ].0 ,r.[ holds
(((#Z 1) ^ ) `| ].0 ,r.[) . x0 = (((- 1) (#) ((#Z 2) ^ )) | ].0 ,r.[) . x0

proof

let x0 be Real; :: thesis: ( x0 in ].0 ,r.[ implies (((#Z 1) ^ ) `| ].0 ,r.[) . x0 = (((- 1) (#) ((#Z 2) ^ )) | ].0 ,r.[) . x0 )
assume A8: x0 in ].0 ,r.[ ; :: thesis: (((#Z 1) ^ ) `| ].0 ,r.[) . x0 = (((- 1) (#) ((#Z 2) ^ )) | ].0 ,r.[) . x0
A9: (#Z 1) ^ is_differentiable_on ].0 ,r.[ by Th34;
A11: x0 <> 0 by A8, XXREAL_1:4;
A13: dom ((- 1) (#) ((#Z 2) ^ )) = dom ((#Z (1 + 1)) ^ ) by VALUED_1:def 5;
(((#Z 1) ^ ) `| ].0 ,r.[) . x0 = diff ((#Z 1) ^ ),x0 by A9, A8, FDIFF_1:def 8
.= - (1 / ((x0 #Z 1) ^2 )) by ThA, A11
.= - (1 / ((x0 |^ 1) ^2 )) by PREPOWER:46
.= - (1 / (x0 |^ (1 + 1))) by NEWTON:13
.= - (1 / (x0 #Z 2)) by PREPOWER:46
.= - (1 / ((#Z 2) . x0)) by TAYLOR_1:def 1
.= - (1 * (((#Z 2) . x0) " )) by XCMPLX_0:def 9
.= - (1 * (((#Z 2) ^ ) . x0)) by A8, A2, RFUNCT_1:def 8
.= (- 1) * (((#Z 2) ^ ) . x0)
.= ((- 1) (#) ((#Z 2) ^ )) . x0 by A8, A2, A13, VALUED_1:def 5
.= (((- 1) (#) ((#Z 2) ^ )) | ].0 ,r.[) . x0 by A8, FUNCT_1:72 ;
hence (((#Z 1) ^ ) `| ].0 ,r.[) . x0 = (((- 1) (#) ((#Z 2) ^ )) | ].0 ,r.[) . x0 ; :: thesis: verum

end;

hence ((#Z 1) ^ ) `| ].0 ,r.[ = ((- 1) (#) ((#Z 2) ^ )) | ].0 ,r.[ by A5, A4, PARTFUN1:34; :: thesis: verum