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

theorem Th56:
  cos|[.PI+2*PI*i,2*PI+2*PI*i.] is increasing
proof
  defpred P[Integer] means cos|[.PI+T($1),2*PI+T($1).] is increasing;
A1: for i holds P[i] implies P[i-1] & P[i+1]
  proof
    let i such that
A2: P[i];
    set Z = [.PI+T(i-1+1),2*PI+T(i-1+1).];
    thus P[i-1]
    proof
      set Y = [.PI+T(i-1),2*PI+T(i-1).];
      now
        let r1, r2;
        assume r1 in Y /\ dom cos & r2 in Y /\ dom cos;
        then
A3:     r1+2*PI in Z /\ dom cos & r2+2*PI in Z /\ dom cos by 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,A3,RFUNCT_2:20;
        then cos.(r1+2*PI*1) < cos.r2 by Th10;
        hence cos.r1 < cos.r2 by Th10;
      end;
      hence thesis by RFUNCT_2:20;
    end;
    set Y = [.PI+T(i+1),2*PI+T(i+1).];
A4: Z = [.PI+T(i+1-1),2*PI+T(i+1-1).];
    now
      let r1, r2;
      assume r1 in Y /\ dom cos & r2 in Y /\ dom cos;
      then
A5:   r1-2*PI in Z /\ dom cos & r2-2*PI in Z /\ dom cos by A4,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,A5,RFUNCT_2:20;
      then cos.(r1+2*PI*(-1)) < cos.r2 by Th10;
      hence cos.r1 < cos.r2 by Th10;
    end;
    hence thesis by RFUNCT_2:20;
  end;
A6: P[0] by COMPTRIG:26;
  for i holds P[i] from INT_1:sch 4(A6,A1);
  hence thesis;
end;
