let n be non empty Element of NAT ; :: thesis: for i being Element of NAT
for f being PartFunc of (REAL-NS n),(REAL-NS 1)
for g being PartFunc of (REAL n),REAL
for x being Point of (REAL-NS n)
for y being Element of REAL n st f = <>* g & x = y & f is_partial_differentiable_in x,i holds
(partdiff f,x,i) . <*1*> = <*(partdiff g,y,i)*>
let i be Element of NAT ; :: thesis: for f being PartFunc of (REAL-NS n),(REAL-NS 1)
for g being PartFunc of (REAL n),REAL
for x being Point of (REAL-NS n)
for y being Element of REAL n st f = <>* g & x = y & f is_partial_differentiable_in x,i holds
(partdiff f,x,i) . <*1*> = <*(partdiff g,y,i)*>
let f be PartFunc of (REAL-NS n),(REAL-NS 1); :: thesis: for g being PartFunc of (REAL n),REAL
for x being Point of (REAL-NS n)
for y being Element of REAL n st f = <>* g & x = y & f is_partial_differentiable_in x,i holds
(partdiff f,x,i) . <*1*> = <*(partdiff g,y,i)*>
let g be PartFunc of (REAL n),REAL ; :: thesis: for x being Point of (REAL-NS n)
for y being Element of REAL n st f = <>* g & x = y & f is_partial_differentiable_in x,i holds
(partdiff f,x,i) . <*1*> = <*(partdiff g,y,i)*>
let x be Point of (REAL-NS n); :: thesis: for y being Element of REAL n st f = <>* g & x = y & f is_partial_differentiable_in x,i holds
(partdiff f,x,i) . <*1*> = <*(partdiff g,y,i)*>
let y be Element of REAL n; :: thesis: ( f = <>* g & x = y & f is_partial_differentiable_in x,i implies (partdiff f,x,i) . <*1*> = <*(partdiff g,y,i)*> )
assume that
A1: f = <>* g and
A2: x = y and
A3: f is_partial_differentiable_in x,i ; :: thesis: (partdiff f,x,i) . <*1*> = <*(partdiff g,y,i)*>
A4: f * (reproj i,x) is_differentiable_in (Proj i,n) . x by A3, Def9;
A5: <*((proj i,n) . y)*> = (Proj i,n) . x by A2, Def4;
f * (reproj i,x) = <>* (g * (reproj i,y)) by A1, A2, Th13;
hence (partdiff f,x,i) . <*1*> = <*(partdiff g,y,i)*> by A4, A5, Th10; :: thesis: verum