let n be Nat; :: thesis: for m being Nat of n
for RAS being ReperAlgebra of n
for a being Point of RAS
for p being Tuple of (n + 1),RAS
for W being ATLAS of RAS
for x being Tuple of (n + 1),W st (a,x) . W = p & m <= n holds
(a,(x +* ((m + 1),(x . m)))) . W = p +* ((m + 1),(p . m))
let m be Nat of n; :: thesis: for RAS being ReperAlgebra of n
for a being Point of RAS
for p being Tuple of (n + 1),RAS
for W being ATLAS of RAS
for x being Tuple of (n + 1),W st (a,x) . W = p & m <= n holds
(a,(x +* ((m + 1),(x . m)))) . W = p +* ((m + 1),(p . m))
let RAS be ReperAlgebra of n; :: thesis: for a being Point of RAS
for p being Tuple of (n + 1),RAS
for W being ATLAS of RAS
for x being Tuple of (n + 1),W st (a,x) . W = p & m <= n holds
(a,(x +* ((m + 1),(x . m)))) . W = p +* ((m + 1),(p . m))
let a be Point of RAS; :: thesis: for p being Tuple of (n + 1),RAS
for W being ATLAS of RAS
for x being Tuple of (n + 1),W st (a,x) . W = p & m <= n holds
(a,(x +* ((m + 1),(x . m)))) . W = p +* ((m + 1),(p . m))
let p be Tuple of (n + 1),RAS; :: thesis: for W being ATLAS of RAS
for x being Tuple of (n + 1),W st (a,x) . W = p & m <= n holds
(a,(x +* ((m + 1),(x . m)))) . W = p +* ((m + 1),(p . m))
let W be ATLAS of RAS; :: thesis: for x being Tuple of (n + 1),W st (a,x) . W = p & m <= n holds
(a,(x +* ((m + 1),(x . m)))) . W = p +* ((m + 1),(p . m))
let x be Tuple of (n + 1),W; :: thesis: ( (a,x) . W = p & m <= n implies (a,(x +* ((m + 1),(x . m)))) . W = p +* ((m + 1),(p . m)) )
assume that
A1: (a,x) . W = p and
A2: m <= n ; :: thesis: (a,(x +* ((m + 1),(x . m)))) . W = p +* ((m + 1),(p . m))
W . (a,p) = x by A1, Th15;
then W . (a,(p +* ((m + 1),(p . m)))) = x +* ((m + 1),(x . m)) by A2, Th31;
hence (a,(x +* ((m + 1),(x . m)))) . W = p +* ((m + 1),(p . m)) by Th15; :: thesis: verum