reserve x,X for set,
        r,r1,r2,s for Real,
        i,j,k,m,n for Nat;
reserve p,q for Point of TOP-REAL n;

theorem Th22:
  1 <= i & i < j & j <= n implies
    (Mx2Tran Rotation(i,j,n,r)).p.j=p.i*(sin r)+p.j*(cos r)
proof
  set O=Rotation(i,j,n,r),M=Mx2Tran O,Mp=M.p,S=Seg n;
  assume that
   A1: 1<=i & i<j and
   A2: j<=n;
  1<=j by A1,XXREAL_0:2;
  hence Mp.j=@p"*"Col(O,j) by A2,MATRTOP1:18
   .=p.i*(sin r)+p.j*(cos r) by A1,A2,Th16;
end;
