reserve x for Real;

theorem Th26:
  cos|[.PI,2*PI.] is increasing
proof
  now
    let r1,r2 be Real;
    assume that
A1: r1 in [.PI,2*PI.] /\ dom cos and
A2: r2 in [.PI,2*PI.] /\ dom cos and
A3: r1 < r2;
A4: r1 in dom cos by A1,XBOOLE_0:def 4;
    set r3 = (r1+r2)/2;
    r1 in [.PI,2*PI.] by A1,XBOOLE_0:def 4;
    then
A5: PI <= r1 by XXREAL_1:1;
    |.cos r3.| <= 1 by SIN_COS:27;
    then
A6: |.cos.r3.| <= 1 by SIN_COS:def 19;
    then
A7: cos.r3 <= 1 by ABSVALUE:5;
    r2 in [.PI,2*PI.] by A2,XBOOLE_0:def 4;
    then
A8: r2 <= 2*PI by XXREAL_1:1;
A9: r1 < r3 by A3,XREAL_1:226;
    then
A10: PI < r3 by A5,XXREAL_0:2;
A11: r3 < r2 by A3,XREAL_1:226;
    then r3 < 2*PI by A8,XXREAL_0:2;
    then r3 in ].PI,2*PI.[ by A10,XXREAL_1:4;
    then
A12: r3 in ].PI,2*PI.[ /\ dom cos by SIN_COS:24,XBOOLE_0:def 4;
    |.cos r2.| <= 1 by SIN_COS:27;
    then |.cos.r2.| <= 1 by SIN_COS:def 19;
    then
A13: cos.r2 >= -1 by ABSVALUE:5;
A14: r2 in dom cos by A2,XBOOLE_0:def 4;
A15: cos.r3 >= -1 by A6,ABSVALUE:5;
    now
      per cases by A5,XXREAL_0:1;
      suppose
A16:    PI < r1;
        then
A17:    PI < r2 by A3,XXREAL_0:2;
        now
          per cases by A8,XXREAL_0:1;
          suppose
A18:        r2 < 2*PI;
            then r1 < 2*PI by A3,XXREAL_0:2;
            then r1 in ].PI,2*PI.[ by A16,XXREAL_1:4;
            then
A19:        r1 in ].PI,2*PI.[ /\ dom cos by A4,XBOOLE_0:def 4;
            r2 in ].PI,2*PI.[ by A17,A18,XXREAL_1:4;
            then r2 in ].PI,2*PI.[ /\ dom cos by A14,XBOOLE_0:def 4;
            hence cos.r2 > cos.r1 by A3,A19,Th22,RFUNCT_2:20;
          end;
          suppose
A20:        r2 = 2*PI;
            then r1 in ].PI,2*PI.[ by A3,A16,XXREAL_1:4;
            then r1 in ].PI,2*PI.[ /\ dom cos by A4,XBOOLE_0:def 4;
            then
A21:        cos.r3 > cos.r1 by A9,A12,Th22,RFUNCT_2:20;
            assume cos.r2 <= cos.r1;
            hence contradiction by A7,A20,A21,SIN_COS:76,XXREAL_0:2;
          end;
        end;
        hence cos.r2 > cos.r1;
      end;
      suppose
A22:    PI = r1;
        now
          per cases by A8,XXREAL_0:1;
          suppose
            r2 < 2*PI;
            then r2 in ].PI,2*PI.[ by A3,A22,XXREAL_1:4;
            then r2 in ].PI,2*PI.[ /\ dom cos by A14,XBOOLE_0:def 4;
            then
A23:        cos.r2 > cos.r3 by A11,A12,Th22,RFUNCT_2:20;
            assume cos.r2 <= cos.r1;
            hence contradiction by A15,A13,A22,A23,SIN_COS:76,XXREAL_0:1;
          end;
          suppose
            r2 = 2*PI;
            hence cos.r2 > cos.r1 by A22,SIN_COS:76;
          end;
        end;
        hence cos.r2 > cos.r1;
      end;
    end;
    hence cos.r2 > cos.r1;
  end;
  hence thesis by RFUNCT_2:20;
end;
