let n be Nat; :: thesis: for RAS being ReperAlgebra of n

for a, b being Point of RAS

for p being Tuple of (n + 1),RAS

for W being ATLAS of RAS

for v being Vector of W

for x being Tuple of (n + 1),W st (a,x) . W = p & (a,v) . W = b & *' (a,p) = b holds

Phi x = v

let RAS be ReperAlgebra of n; :: thesis: for a, b being Point of RAS

for p being Tuple of (n + 1),RAS

for W being ATLAS of RAS

for v being Vector of W

for x being Tuple of (n + 1),W st (a,x) . W = p & (a,v) . W = b & *' (a,p) = b holds

Phi x = v

let a, b be Point of RAS; :: thesis: for p being Tuple of (n + 1),RAS

for W being ATLAS of RAS

for v being Vector of W

for x being Tuple of (n + 1),W st (a,x) . W = p & (a,v) . W = b & *' (a,p) = b holds

Phi x = v

let p be Tuple of (n + 1),RAS; :: thesis: for W being ATLAS of RAS

for v being Vector of W

for x being Tuple of (n + 1),W st (a,x) . W = p & (a,v) . W = b & *' (a,p) = b holds

Phi x = v

let W be ATLAS of RAS; :: thesis: for v being Vector of W

for x being Tuple of (n + 1),W st (a,x) . W = p & (a,v) . W = b & *' (a,p) = b holds

Phi x = v

let v be Vector of W; :: thesis: for x being Tuple of (n + 1),W st (a,x) . W = p & (a,v) . W = b & *' (a,p) = b holds

Phi x = v

let x be Tuple of (n + 1),W; :: thesis: ( (a,x) . W = p & (a,v) . W = b & *' (a,p) = b implies Phi x = v )

assume ( (a,x) . W = p & (a,v) . W = b & *' (a,p) = b ) ; :: thesis: Phi x = v

then Phi (a,x) = v by MIDSP_2:33;

hence Phi x = v by Def12; :: thesis: verum

for a, b being Point of RAS

for p being Tuple of (n + 1),RAS

for W being ATLAS of RAS

for v being Vector of W

for x being Tuple of (n + 1),W st (a,x) . W = p & (a,v) . W = b & *' (a,p) = b holds

Phi x = v

let RAS be ReperAlgebra of n; :: thesis: for a, b being Point of RAS

for p being Tuple of (n + 1),RAS

for W being ATLAS of RAS

for v being Vector of W

for x being Tuple of (n + 1),W st (a,x) . W = p & (a,v) . W = b & *' (a,p) = b holds

Phi x = v

let a, b be Point of RAS; :: thesis: for p being Tuple of (n + 1),RAS

for W being ATLAS of RAS

for v being Vector of W

for x being Tuple of (n + 1),W st (a,x) . W = p & (a,v) . W = b & *' (a,p) = b holds

Phi x = v

let p be Tuple of (n + 1),RAS; :: thesis: for W being ATLAS of RAS

for v being Vector of W

for x being Tuple of (n + 1),W st (a,x) . W = p & (a,v) . W = b & *' (a,p) = b holds

Phi x = v

let W be ATLAS of RAS; :: thesis: for v being Vector of W

for x being Tuple of (n + 1),W st (a,x) . W = p & (a,v) . W = b & *' (a,p) = b holds

Phi x = v

let v be Vector of W; :: thesis: for x being Tuple of (n + 1),W st (a,x) . W = p & (a,v) . W = b & *' (a,p) = b holds

Phi x = v

let x be Tuple of (n + 1),W; :: thesis: ( (a,x) . W = p & (a,v) . W = b & *' (a,p) = b implies Phi x = v )

assume ( (a,x) . W = p & (a,v) . W = b & *' (a,p) = b ) ; :: thesis: Phi x = v

then Phi (a,x) = v by MIDSP_2:33;

hence Phi x = v by Def12; :: thesis: verum