deffunc H_{1}() -> set = {0,1,2};

deffunc H_{2}( Element of H_{1}(), Element of H_{1}()) -> Element of {0,1,2} = OpEx2 ($1,$2);

ex f being BinOp of H_{1}() st

for x, y being Element of H_{1}() holds f . (x,y) = H_{2}(x,y)
from BINOP_1:sch 4();

then consider f being BinOp of H_{1}() such that

A1: for x, y being Element of H_{1}() holds f . (x,y) = H_{2}(x,y)
;

take f ; :: thesis: for x, y being Element of {0,1,2} holds

( ( ( x = 1 or y = 1 ) implies f . (x,y) = 1 ) & ( x <> 1 & y <> 1 implies f . (x,y) = min (x,y) ) )

let x, y be Element of {0,1,2}; :: thesis: ( ( ( x = 1 or y = 1 ) implies f . (x,y) = 1 ) & ( x <> 1 & y <> 1 implies f . (x,y) = min (x,y) ) )

f . (x,y) = OpEx2 (x,y) by A1;

hence f . (x,y) = min (x,y) by OpEx2Def, A2; :: thesis: verum

deffunc H

ex f being BinOp of H

for x, y being Element of H

then consider f being BinOp of H

A1: for x, y being Element of H

take f ; :: thesis: for x, y being Element of {0,1,2} holds

( ( ( x = 1 or y = 1 ) implies f . (x,y) = 1 ) & ( x <> 1 & y <> 1 implies f . (x,y) = min (x,y) ) )

let x, y be Element of {0,1,2}; :: thesis: ( ( ( x = 1 or y = 1 ) implies f . (x,y) = 1 ) & ( x <> 1 & y <> 1 implies f . (x,y) = min (x,y) ) )

hereby :: thesis: ( x <> 1 & y <> 1 implies f . (x,y) = min (x,y) )

assume A2:
( x <> 1 & y <> 1 )
; :: thesis: f . (x,y) = min (x,y)assume A2:
( x = 1 or y = 1 )
; :: thesis: f . (x,y) = 1

f . (x,y) = OpEx2 (x,y) by A1;

hence f . (x,y) = 1 by A2, OpEx2Def; :: thesis: verum

end;f . (x,y) = OpEx2 (x,y) by A1;

hence f . (x,y) = 1 by A2, OpEx2Def; :: thesis: verum

f . (x,y) = OpEx2 (x,y) by A1;

hence f . (x,y) = min (x,y) by OpEx2Def, A2; :: thesis: verum