reserve r,t for Real;
reserve i for Integer;
reserve k,n for Nat;
reserve p for Polynomial of F_Real;
reserve e for Element of F_Real;
reserve L for non empty ZeroStr;
reserve z,z0,z1,z2 for Element of L;

theorem Th15:
  0 <= r <= PI/2 & cos r = 1/2 implies r = PI/3
  proof
    set X = [.0,PI/2.];
    set f = cos | X;
    assume that
A1: 0 <= r and
A2: r <= PI/2;
A3: r in X by A1,A2,XXREAL_1:1;
    assume
A4: cos r = 1/2;
A5: dom cos = REAL by FUNCT_2:def 1;
A6: PI/3 <= PI/2 by XREAL_1:76;
    then
A7: PI/3 in X by XXREAL_1:1;
A8: dom f = X by A5,RELAT_1:62;
    then f.r = cos(PI/3) by A1,A2,A4,EUCLID10:14,XXREAL_1:1,FUNCT_1:47
    .= f.(PI/3) by A6,A8,XXREAL_1:1,FUNCT_1:47;
    hence thesis by A3,A7,A8,FUNCT_1:def 4;
  end;
