reserve n for Nat;
reserve i for Integer;
reserve r,s,t for Real;
reserve An,Bn,Cn,Dn for Point of TOP-REAL n;
reserve L1,L2 for Element of line_of_REAL n;
reserve A,B,C for Point of TOP-REAL 2;
reserve D for Point of TOP-REAL 2;
reserve a,b,c,d for Real;

theorem Th69:
  sin(2*s) * cos d = cos (2*t) implies
  (r * cos s)^2 + (r * sin s)^2 - 2 * ( r * cos s) * ( r * sin s) * cos d
    = 2 * r^2 * (sin t)^2
  proof
    assume
A1: sin(2*s) * cos d = cos (2*t);
    (r * cos s)^2 + (r * sin s)^2 - 2*(r*cos s)*(r*sin s)*cos d
      =(r * cos s) * (r * cos s) +(r * sin s)^2 - 2*(r*cos s)*(r*sin s)*(cos d)
         by SQUARE_1:def 1
     .=(r * cos s) * (r * cos s) +(r * sin s)*(r* sin s)
         - 2*(r*cos s)*(r*sin s)*cos d by SQUARE_1:def 1
     .= r * r *(cos s * cos s +sin s * sin s)
          - r * 2 * r * cos s * sin s * cos d
     .= r* r * 1 - r * 2 * r * cos s * sin s * cos d by SIN_COS:29
     .= r^2- (r * r) * 2 * cos s * sin s * cos d by SQUARE_1:def 1
     .= r^2- r^2 * 2 * cos s * sin s * cos d by SQUARE_1:def 1
     .= r^2- r^2 * (2 * sin s * cos s) * cos d
     .= r^2 - r^2 * sin (2*s) * cos d by SIN_COS5:5
     .= r^2-r^2*(cos (2*t)) by A1
     .= r^2 - r^2 * ((cos t)^2-(sin t)^2) by SIN_COS5:7
     .= r^2 * ( 1 - ((cos t)^2-(sin t)^2))
     .= r^2* (((cos t)^2+(sin t)^2)-((cos t)^2-(sin t)^2)) by SIN_COS:29
     .= 2 * r^2 * (sin t)^2;
    hence thesis;
  end;
