let i be integer number ; :: thesis: cos | [.(PI + ((2 * PI ) * i)),((2 * PI ) + ((2 * PI ) * i)).] is increasing
defpred S₁[ Integer] means cos | [.(PI + H₁($1)),((2 * PI ) + H₁($1)).] is increasing ;
A1: S₁[ 0 ] by COMPTRIG:42;
A2: for i being integer number st S₁[i] holds
( S₁[i - 1] & S₁[i + 1] )

proof

let i be integer number ; :: thesis: ( S₁[i] implies ( S₁[i - 1] & S₁[i + 1] ) )
assume A3: S₁[i] ; :: thesis: ( S₁[i - 1] & S₁[i + 1] )
set Z = [.(PI + H₁((i - 1) + 1)),((2 * PI ) + H₁((i - 1) + 1)).];
thus S₁[i - 1] :: thesis: S₁[i + 1]

proof

set Y = [.(PI + H₁(i - 1)),((2 * PI ) + H₁(i - 1)).];

now

let r1, r2 be Element of REAL ; :: thesis: ( r1 in [.(PI + H₁(i - 1)),((2 * PI ) + H₁(i - 1)).] /\ (dom cos ) & r2 in [.(PI + H₁(i - 1)),((2 * PI ) + H₁(i - 1)).] /\ (dom cos ) & r1 < r2 implies cos . r1 < cos . r2 )
assume ( r1 in [.(PI + H₁(i - 1)),((2 * PI ) + H₁(i - 1)).] /\ (dom cos ) & r2 in [.(PI + H₁(i - 1)),((2 * PI ) + H₁(i - 1)).] /\ (dom cos ) ) ; :: thesis: ( r1 < r2 implies cos . r1 < cos . r2 )
then A4: ( r1 + (2 * PI ) in [.(PI + H₁((i - 1) + 1)),((2 * PI ) + H₁((i - 1) + 1)).] /\ (dom cos ) & r2 + (2 * PI ) in [.(PI + H₁((i - 1) + 1)),((2 * PI ) + H₁((i - 1) + 1)).] /\ (dom cos ) ) by Lm2, Lm25;
assume r1 < r2 ; :: thesis: cos . r1 < cos . r2
then r1 + (2 * PI ) < r2 + (2 * PI ) by XREAL_1:8;
then cos . (r1 + (2 * PI )) < cos . (r2 + ((2 * PI ) * 1)) by A3, A4, RFUNCT_2:43;
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_2:43; :: thesis: verum

end;

set Y = [.(PI + H₁(i + 1)),((2 * PI ) + H₁(i + 1)).];
A5: [.(PI + H₁((i - 1) + 1)),((2 * PI ) + H₁((i - 1) + 1)).] = [.(PI + H₁((i + 1) - 1)),((2 * PI ) + H₁((i + 1) - 1)).] ;

now

let r1, r2 be Element of REAL ; :: thesis: ( r1 in [.(PI + H₁(i + 1)),((2 * PI ) + H₁(i + 1)).] /\ (dom cos ) & r2 in [.(PI + H₁(i + 1)),((2 * PI ) + H₁(i + 1)).] /\ (dom cos ) & r1 < r2 implies cos . r1 < cos . r2 )
assume ( r1 in [.(PI + H₁(i + 1)),((2 * PI ) + H₁(i + 1)).] /\ (dom cos ) & r2 in [.(PI + H₁(i + 1)),((2 * PI ) + H₁(i + 1)).] /\ (dom cos ) ) ; :: thesis: ( r1 < r2 implies cos . r1 < cos . r2 )
then A6: ( r1 - (2 * PI ) in [.(PI + H₁((i - 1) + 1)),((2 * PI ) + H₁((i - 1) + 1)).] /\ (dom cos ) & r2 - (2 * PI ) in [.(PI + H₁((i - 1) + 1)),((2 * PI ) + H₁((i - 1) + 1)).] /\ (dom cos ) ) by A5, Lm2, Lm27;
assume r1 < r2 ; :: thesis: cos . r1 < cos . r2
then r1 - (2 * PI ) < r2 - (2 * PI ) by XREAL_1:11;
then cos . (r1 - (2 * PI )) < cos . (r2 + ((2 * PI ) * (- 1))) by A3, A6, RFUNCT_2:43;
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_2:43; :: thesis: verum

end;

for i being integer number holds S₁[i] from INT_1:sch 4(A1, A2);
hence cos | [.(PI + ((2 * PI ) * i)),((2 * PI ) + ((2 * PI ) * i)).] is increasing ; :: thesis: verum