defpred S_{1}[ Element of MSAEnd U1, Element of MSAEnd U1, set ] means $3 = $2 ** $1;

A1: for x, y being Element of MSAEnd U1 ex m being Element of MSAEnd U1 st S_{1}[x,y,m]

for x, y being Element of MSAEnd U1 holds S_{1}[x,y,B . (x,y)]
from BINOP_1:sch 3(A1);

hence ex b_{1} being BinOp of (MSAEnd U1) st

for x, y being Element of MSAEnd U1 holds b_{1} . (x,y) = y ** x
; :: thesis: verum

A1: for x, y being Element of MSAEnd U1 ex m being Element of MSAEnd U1 st S

proof

ex B being BinOp of (MSAEnd U1) st
let x, y be Element of MSAEnd U1; :: thesis: ex m being Element of MSAEnd U1 st S_{1}[x,y,m]

reconsider xx = x, yy = y as ManySortedFunction of U1,U1 ;

reconsider m = yy ** xx as Element of MSAEnd U1 by Th8;

take m ; :: thesis: S_{1}[x,y,m]

thus S_{1}[x,y,m]
; :: thesis: verum

end;reconsider xx = x, yy = y as ManySortedFunction of U1,U1 ;

reconsider m = yy ** xx as Element of MSAEnd U1 by Th8;

take m ; :: thesis: S

thus S

for x, y being Element of MSAEnd U1 holds S

hence ex b

for x, y being Element of MSAEnd U1 holds b