let i be Integer; :: thesis: cos | [.((2 * PI) * i),(PI + ((2 * PI) * i)).] is decreasing
defpred S₁[ Integer] means cos | [.H₁($1),(PI + H₁($1)).] is decreasing ;
A1: for i being Integer st S₁[i] holds
( S₁[i - 1] & S₁[i + 1] )

proof

let i be Integer; :: thesis: ( S₁[i] implies ( S₁[i - 1] & S₁[i + 1] ) )
assume A2: S₁[i] ; :: thesis: ( S₁[i - 1] & S₁[i + 1] )
set Z = [.(0 + H₁((i - 1) + 1)),(PI + H₁((i - 1) + 1)).];
thus S₁[i - 1] :: thesis: S₁[i + 1]

proof

set Y = [.H₁(i - 1),(PI + H₁(i - 1)).];
A3: [.H₁(i - 1),(PI + H₁(i - 1)).] = [.(0 + H₁(i - 1)),(PI + H₁(i - 1)).] ;

now :: thesis: for r1, r2 being Real st r1 in [.H₁(i - 1),(PI + H₁(i - 1)).] /\ (dom cos) & r2 in [.H₁(i - 1),(PI + H₁(i - 1)).] /\ (dom cos) & r1 < r2 holds
cos . r1 > cos . r2

let r1, r2 be Real; :: thesis: ( r1 in [.H₁(i - 1),(PI + H₁(i - 1)).] /\ (dom cos) & r2 in [.H₁(i - 1),(PI + H₁(i - 1)).] /\ (dom cos) & r1 < r2 implies cos . r1 > cos . r2 )
assume ( r1 in [.H₁(i - 1),(PI + H₁(i - 1)).] /\ (dom cos) & r2 in [.H₁(i - 1),(PI + H₁(i - 1)).] /\ (dom cos) ) ; :: thesis: ( r1 < r2 implies cos . r1 > cos . r2 )
then A4: ( r1 + (2 * PI) in [.(0 + H₁((i - 1) + 1)),(PI + H₁((i - 1) + 1)).] /\ (dom cos) & r2 + (2 * PI) in [.(0 + H₁((i - 1) + 1)),(PI + H₁((i - 1) + 1)).] /\ (dom cos) ) by A3, Lm12, SIN_COS:24;
assume r1 < r2 ; :: thesis: cos . r1 > cos . 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; :: thesis: verum

end;

hence S₁[i - 1] by RFUNCT_2:21; :: thesis: verum

end;

set Y = [.H₁(i + 1),(PI + H₁(i + 1)).];
A5: ( [.H₁(i + 1),(PI + H₁(i + 1)).] = [.(0 + H₁(i + 1)),(PI + H₁(i + 1)).] & [.(0 + H₁((i - 1) + 1)),(PI + H₁((i - 1) + 1)).] = [.H₁((i + 1) - 1),(PI + H₁((i + 1) - 1)).] ) ;

now :: thesis: for r1, r2 being Real st r1 in [.H₁(i + 1),(PI + H₁(i + 1)).] /\ (dom cos) & r2 in [.H₁(i + 1),(PI + H₁(i + 1)).] /\ (dom cos) & r1 < r2 holds
cos . r1 > cos . r2

let r1, r2 be Real; :: thesis: ( r1 in [.H₁(i + 1),(PI + H₁(i + 1)).] /\ (dom cos) & r2 in [.H₁(i + 1),(PI + H₁(i + 1)).] /\ (dom cos) & r1 < r2 implies cos . r1 > cos . r2 )
assume ( r1 in [.H₁(i + 1),(PI + H₁(i + 1)).] /\ (dom cos) & r2 in [.H₁(i + 1),(PI + H₁(i + 1)).] /\ (dom cos) ) ; :: thesis: ( r1 < r2 implies cos . r1 > cos . r2 )
then A6: ( r1 - (2 * PI) in [.(0 + H₁((i - 1) + 1)),(PI + H₁((i - 1) + 1)).] /\ (dom cos) & r2 - (2 * PI) in [.(0 + H₁((i - 1) + 1)),(PI + H₁((i - 1) + 1)).] /\ (dom cos) ) by A5, Lm14, SIN_COS:24;
assume r1 < r2 ; :: thesis: cos . r1 > cos . 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; :: thesis: verum

end;

hence S₁[i + 1] by RFUNCT_2:21; :: thesis: verum

end;

A7: S₁[ 0 ] by COMPTRIG:25;
for i being Integer holds S₁[i] from INT_1:sch 4(A7, A1);
hence cos | [.((2 * PI) * i),(PI + ((2 * PI) * i)).] is decreasing ; :: thesis: verum