let n be non zero Nat; :: thesis: for i being 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 holds

( f is_partial_differentiable_in x,i iff g is_partial_differentiable_in y,i )

let i be 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 holds

( f is_partial_differentiable_in x,i iff g is_partial_differentiable_in 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 holds

( f is_partial_differentiable_in x,i iff g is_partial_differentiable_in 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 holds

( f is_partial_differentiable_in x,i iff g is_partial_differentiable_in y,i )

let x be Point of (REAL-NS n); :: thesis: for y being Element of REAL n st f = <>* g & x = y holds

( f is_partial_differentiable_in x,i iff g is_partial_differentiable_in y,i )

let y be Element of REAL n; :: thesis: ( f = <>* g & x = y implies ( f is_partial_differentiable_in x,i iff g is_partial_differentiable_in y,i ) )

assume that

A1: f = <>* g and

A2: x = y ; :: thesis: ( f is_partial_differentiable_in x,i iff g is_partial_differentiable_in y,i )

A3: <*((proj (i,n)) . y)*> = (Proj (i,n)) . x by A2, Def4;

f * (reproj (i,x)) = <>* (g * (reproj (i,y))) by A1, A2, Th13;

hence ( f is_partial_differentiable_in x,i iff g is_partial_differentiable_in y,i ) by A3, Th7, Th8; :: thesis: verum

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 holds

( f is_partial_differentiable_in x,i iff g is_partial_differentiable_in y,i )

let i be 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 holds

( f is_partial_differentiable_in x,i iff g is_partial_differentiable_in 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 holds

( f is_partial_differentiable_in x,i iff g is_partial_differentiable_in 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 holds

( f is_partial_differentiable_in x,i iff g is_partial_differentiable_in y,i )

let x be Point of (REAL-NS n); :: thesis: for y being Element of REAL n st f = <>* g & x = y holds

( f is_partial_differentiable_in x,i iff g is_partial_differentiable_in y,i )

let y be Element of REAL n; :: thesis: ( f = <>* g & x = y implies ( f is_partial_differentiable_in x,i iff g is_partial_differentiable_in y,i ) )

assume that

A1: f = <>* g and

A2: x = y ; :: thesis: ( f is_partial_differentiable_in x,i iff g is_partial_differentiable_in y,i )

A3: <*((proj (i,n)) . y)*> = (Proj (i,n)) . x by A2, Def4;

f * (reproj (i,x)) = <>* (g * (reproj (i,y))) by A1, A2, Th13;

hence ( f is_partial_differentiable_in x,i iff g is_partial_differentiable_in y,i ) by A3, Th7, Th8; :: thesis: verum