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)
proof
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)
per cases ( ( x2 = 1 & y = 0 ) or x2 <> 1 or y <> 0 ) ;
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;
suppose W1: x2 <> 1 ; :: thesis: I_{1} . (x1,y) >= I_{1} . (x2,y)
x2 <= 1 by XXREAL_1:1;
then Y3: x2 < 1 by ;
then Y2: I_{1} . (x2,y) = 1 by I1Impl;
x1 < 1 by ;
hence I_{1} . (x1,y) >= I_{1} . (x2,y) by ; :: thesis: verum
end;
suppose W2: y <> 0 ; :: thesis: I_{1} . (x1,y) >= I_{1} . (x2,y)
y4: y >= 0 by XXREAL_1:1;
I_{1} . (x1,y) = 1 by ;
hence I_{1} . (x1,y) >= I_{1} . (x2,y) by XXREAL_1:1; :: thesis: verum
end;
end;
end;
b2: for x, y1, y2 being Element of [.0,1.] st y1 <= y2 holds
I_{1} . (x,y1) <= I_{1} . (x,y2)
proof
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)
per cases ( ( y1 = 0 & x = 1 ) or y1 <> 0 or x <> 1 ) ;
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;
suppose y1 <> 0 ; :: thesis: I_{1} . (x,y1) <= I_{1} . (x,y2)
then A2: y1 > 0 by XXREAL_1:1;
then I_{1} . (x,y1) = 1 by I1Impl;
hence I_{1} . (x,y1) <= I_{1} . (x,y2) by ; :: thesis: verum
end;
suppose B1: x <> 1 ; :: thesis: I_{1} . (x,y1) <= I_{1} . (x,y2)
x <= 1 by XXREAL_1:1;
then A2: x < 1 by ;
then I_{1} . (x,y1) = 1 by I1Impl;
hence I_{1} . (x,y1) <= I_{1} . (x,y2) by ; :: thesis: verum
end;
end;
end;
( 0 in [.0,1.] & 1 in [.0,1.] ) by XXREAL_1:1;
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 ; :: thesis: verum