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 Th25:
  1 <= i & i < j & j <= n & s^2 <= (p.i)^2+(p.j)^2 implies
    ex r st (Mx2Tran Rotation(i,j,n,r)).p.j = s
proof
  set pk=p.i,pj=p.j,pkj=pk^2+pj^2,ps=pkj-s^2;
  assume that
   A1: 1<=i & i<j & j<=n and
   A2: s^2<=pk^2+pj^2;
  0<=s*s by XREAL_1:63;
  then A3: ps<=pkj-0 by XREAL_1:6;
  A4: s^2-s^2<=ps by A2,XREAL_1:6;
  then (sqrt ps)^2=ps by SQUARE_1:def 2;
  then consider r such that
   A5: (Mx2Tran Rotation(i,j,n,r)).p.i=sqrt ps by A1,A3,Th24;
  set M=Mx2Tran Rotation(i,j,n,r),Mp=M.p;
  pkj=(sqrt ps)^2+(Mp.j)*(Mp.j) by A5,A1,Lm6
   .=ps+(Mp.j)*(Mp.j) by A4,SQUARE_1:def 2;
  then A6: s^2=(Mp.j)^2;
  per cases by A6,SQUARE_1:40;
  suppose Mp.j=s;
    hence thesis;
  end;
  suppose A7: Mp.j=-s;
    take R=r+PI;
    thus(Mx2Tran Rotation(i,j,n,R)).p.j=p.i*(sin R)+p.j*(cos R) by A1,Th22
    .=p.i*(-sin r)+p.j*(cos R) by SIN_COS:79
    .=p.i*(-sin r)+p.j*(-cos r) by SIN_COS:79
    .=-(p.i*(sin r)+p.j*(cos r))
    .=-Mp.j by A1,Th22
    .=s by A7;
  end;
end;
