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 Th21:
  1 <= i & i < j & j <= n implies
    (Mx2Tran Rotation(i,j,n,r)).p.i=p.i*(cos r)+p.j*(-sin r)
proof
  set O=Rotation(i,j,n,r),M=Mx2Tran O,Mp=M.p,S=Seg n;
  assume that
  A1: 1<=i and
  A2: i<j & j<=n;
  i<=n by A2,XXREAL_0:2;
  hence Mp.i=@p"*"Col(O,i) by A1,MATRTOP1:18
  .=p.i*(cos r)+p.j*(-sin r) by A1,A2,Th15;
end;
