let r be Real; :: thesis: for i, j, n being Nat
for p being Point of (TOP-REAL n) st 1 <= i & i < j & j <= n holds
((Mx2Tran (Rotation (i,j,n,r))) . p) . j = ((p . i) * (sin r)) + ((p . j) * (cos r))
let i, j, n be Nat; :: thesis: for p being Point of (TOP-REAL n) st 1 <= i & i < j & j <= n holds
((Mx2Tran (Rotation (i,j,n,r))) . p) . j = ((p . i) * (sin r)) + ((p . j) * (cos r))
let p be Point of (TOP-REAL n); :: thesis: ( 1 <= i & i < j & j <= n implies ((Mx2Tran (Rotation (i,j,n,r))) . p) . j = ((p . i) * (sin r)) + ((p . j) * (cos r)) )
set O = Rotation (i,j,n,r);
set M = Mx2Tran (Rotation (i,j,n,r));
set Mp = (Mx2Tran (Rotation (i,j,n,r))) . p;
set S = Seg n;
assume that
A1: ( 1 <= i & i < j ) and
A2: j <= n ; :: thesis: ((Mx2Tran (Rotation (i,j,n,r))) . p) . j = ((p . i) * (sin r)) + ((p . j) * (cos r))
1 <= j by A1, XXREAL_0:2;
hence ((Mx2Tran (Rotation (i,j,n,r))) . p) . j = (@ p) "*" (Col ((Rotation (i,j,n,r)),j)) by A2, MATRTOP1:18
.= ((p . i) * (sin r)) + ((p . j) * (cos r)) by A1, A2, Th16 ;
:: thesis: verum