set f = I_{1} ;

b1: for x1, x2, y being Element of [.0,1.] st x1 <= x2 holds

I_{1} . (x1,y) >= I_{1} . (x2,y)

I_{1} . (x,y1) <= I_{1} . (x,y2)

hence ( I_{1} is decreasing_on_1st & I_{1} is increasing_on_2nd & I_{1} is 00-dominant & I_{1} is 11-dominant & I_{1} is 10-weak ) by b1, b2, I1Impl; :: thesis: verum

b1: for x1, x2, y being Element of [.0,1.] st x1 <= x2 holds

I_{1} . (x1,y) >= I_{1} . (x2,y)

proof

b2:
for x, y1, y2 being Element of [.0,1.] st y1 <= y2 holds
let x1, x2, y be Element of [.0,1.]; :: thesis: ( x1 <= x2 implies I_{1} . (x1,y) >= I_{1} . (x2,y) )

assume A0: x1 <= x2 ; :: thesis: I_{1} . (x1,y) >= I_{1} . (x2,y)

end;assume A0: x1 <= x2 ; :: thesis: I_{1} . (x1,y) >= I_{1} . (x2,y)

per cases
( ( x2 = 1 & y = 0 ) or x2 <> 1 or y <> 0 )
;

end;

suppose
( x2 = 1 & y = 0 )
; :: thesis: I_{1} . (x1,y) >= I_{1} . (x2,y)

then
I_{1} . (x2,y) = 0
by I1Impl;

hence I_{1} . (x1,y) >= I_{1} . (x2,y) by XXREAL_1:1; :: thesis: verum

end;hence I_{1} . (x1,y) >= I_{1} . (x2,y) by XXREAL_1:1; :: thesis: verum

I_{1} . (x,y1) <= I_{1} . (x,y2)

proof

( 0 in [.0,1.] & 1 in [.0,1.] )
by XXREAL_1:1;
let x, y1, y2 be Element of [.0,1.]; :: thesis: ( y1 <= y2 implies I_{1} . (x,y1) <= I_{1} . (x,y2) )

assume A0: y1 <= y2 ; :: thesis: I_{1} . (x,y1) <= I_{1} . (x,y2)

end;assume A0: y1 <= y2 ; :: thesis: I_{1} . (x,y1) <= I_{1} . (x,y2)

per cases
( ( y1 = 0 & x = 1 ) or y1 <> 0 or x <> 1 )
;

end;

suppose
( y1 = 0 & x = 1 )
; :: thesis: I_{1} . (x,y1) <= I_{1} . (x,y2)

then
I_{1} . (x,y1) = 0
by I1Impl;

hence I_{1} . (x,y1) <= I_{1} . (x,y2) by XXREAL_1:1; :: thesis: verum

end;hence I_{1} . (x,y1) <= I_{1} . (x,y2) by XXREAL_1:1; :: thesis: verum

hence ( I_{1} is decreasing_on_1st & I_{1} is increasing_on_2nd & I_{1} is 00-dominant & I_{1} is 11-dominant & I_{1} is 10-weak ) by b1, b2, I1Impl; :: thesis: verum