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 W . (a,p) = x & m <= n holds
W . (a,(p +* ((m + 1),(p . m)))) = x +* ((m + 1),(x . 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 W . (a,p) = x & m <= n holds
W . (a,(p +* ((m + 1),(p . m)))) = x +* ((m + 1),(x . 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 W . (a,p) = x & m <= n holds
W . (a,(p +* ((m + 1),(p . m)))) = x +* ((m + 1),(x . 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 W . (a,p) = x & m <= n holds
W . (a,(p +* ((m + 1),(p . m)))) = x +* ((m + 1),(x . 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 W . (a,p) = x & m <= n holds
W . (a,(p +* ((m + 1),(p . m)))) = x +* ((m + 1),(x . m))
let W be ATLAS of RAS; :: thesis: for x being Tuple of (n + 1),W st W . (a,p) = x & m <= n holds
W . (a,(p +* ((m + 1),(p . m)))) = x +* ((m + 1),(x . m))
let x be Tuple of (n + 1),W; :: thesis: ( W . (a,p) = x & m <= n implies W . (a,(p +* ((m + 1),(p . m)))) = x +* ((m + 1),(x . m)) )
assume that
A1: W . (a,p) = x and
A2: m <= n ; :: thesis: W . (a,(p +* ((m + 1),(p . m)))) = x +* ((m + 1),(x . m))
reconsider m9 = m + 1 as Nat of n by A2, Th8;
set y = W . (a,(p +* (m9,(p . m))));
set z = x +* (m9,(x . m));
for k being Nat of n holds (W . (a,(p +* (m9,(p . m))))) . k = (x +* (m9,(x . m))) . k

proof

let k be Nat of n; :: thesis: (W . (a,(p +* (m9,(p . m))))) . k = (x +* (m9,(x . m))) . k

now :: thesis: (W . (a,(p +* (m9,(p . m))))) . k = (x +* (m9,(x . m))) . k

per cases ( k = m9 or k <> m9 ) ;

suppose A3: k = m9 ; :: thesis: (W . (a,(p +* (m9,(p . m))))) . k = (x +* (m9,(x . m))) . k

thus (W . (a,(p +* (m9,(p . m))))) . k = W . (a,((p +* (m9,(p . m))) . k)) by Def9
.= W . (a,(p . m)) by A3, Th10
.= x . m by A1, Def9
.= (x +* (m9,(x . m))) . k by A3, Th13 ; :: thesis: verum

end;

suppose A4: k <> m9 ; :: thesis: (W . (a,(p +* (m9,(p . m))))) . k = (x +* (m9,(x . m))) . k

thus (W . (a,(p +* (m9,(p . m))))) . k = W . (a,((p +* (m9,(p . m))) . k)) by Def9
.= W . (a,(p . k)) by A4, FUNCT_7:32
.= x . k by A1, Def9
.= (x +* (m9,(x . m))) . k by A4, FUNCT_7:32 ; :: thesis: verum

end;

end;

end;

hence (W . (a,(p +* (m9,(p . m))))) . k = (x +* (m9,(x . m))) . k ; :: thesis: verum

end;

hence W . (a,(p +* ((m + 1),(p . m)))) = x +* ((m + 1),(x . m)) by Th14; :: thesis: verum