 reserve a,b,r for Real;
 reserve A for non empty set;
 reserve X,x for set;
 reserve f,g,F,G for PartFunc of REAL,REAL;
 reserve n for Element of NAT;

theorem
  for x,y be PartFunc of REAL,REAL,
      Z be open Subset of REAL
   st x is differentiable
    & y is differentiable
    & ['0,PI'] c= Z & Z c= dom x & Z c= dom y
    & x`|Z is continuous
    & y`|Z is continuous
    & (for t be Real st t in Z
       holds ((x`|Z).t)^2 + ((y`|Z).t)^2 = 1)
    & y.0 = 0 & y.PI = 0
  holds
    integral(y(#)(x`|Z),0,PI) <= 1/2 * PI
  & (integral(y(#)(x`|Z),0,PI) = 1/2 * PI
        iff
     ((for t be Real st t in [.0,PI.]
       holds y.t = sin.t & x.t = -cos.t + cos.0 + x.0)
          or
      (for t be Real st t in [.0,PI.]
       holds y.t = -sin.t & x.t = cos.t - cos.0 + x.0))) by Lm27,Lm29;
