reserve r,r1,r2, s,x for Real,
  i for Integer;

theorem Th55:
  cos|[.2*PI*i,PI+2*PI*i.] is decreasing
proof
  defpred P[Integer] means cos|[.T($1),PI+T($1).] is decreasing;
A1: for i holds P[i] implies P[i-1] & P[i+1]
  proof
    let i such that
A2: P[i];
    set Z = [.0+T(i-1+1),PI+T(i-1+1).];
    thus P[i-1]
    proof
      set Y = [.T(i-1),PI+T(i-1).];
A3:   Y = [.0+T(i-1),PI+T(i-1).];
      now
        let r1, r2;
        assume r1 in Y /\ dom cos & r2 in Y /\ dom cos;
        then
A4:     r1+2*PI in Z /\ dom cos & r2+2*PI in Z /\ dom cos by A3,Lm12,SIN_COS:24
;
        assume r1 < r2;
        then r1+2*PI < r2+2*PI by XREAL_1:6;
        then cos.(r1+2*PI) > cos.(r2+2*PI*1) by A2,A4,RFUNCT_2:21;
        then cos.(r1+2*PI*1) > cos.r2 by Th10;
        hence cos.r1 > cos.r2 by Th10;
      end;
      hence thesis by RFUNCT_2:21;
    end;
    set Y = [.T(i+1),PI+T(i+1).];
A5: Y = [.0+T(i+1),PI+T(i+1).] & Z = [.T(i+1-1),PI+T(i+1-1).];
    now
      let r1, r2;
      assume r1 in Y /\ dom cos & r2 in Y /\ dom cos;
      then
A6:   r1-2*PI in Z /\ dom cos & r2-2*PI in Z /\ dom cos by A5,Lm14,SIN_COS:24;
      assume r1 < r2;
      then r1-2*PI < r2-2*PI by XREAL_1:9;
      then cos.(r1-2*PI) > cos.(r2+2*PI*(-1)) by A2,A6,RFUNCT_2:21;
      then cos.(r1+2*PI*(-1)) > cos.r2 by Th10;
      hence cos.r1 > cos.r2 by Th10;
    end;
    hence thesis by RFUNCT_2:21;
  end;
A7: P[0] by COMPTRIG:25;
  for i holds P[i] from INT_1:sch 4(A7,A1);
  hence thesis;
end;
