let F be non degenerated right_complementable Abelian add-associative right_zeroed distributive Field-like doubleLoopStr ; :: thesis: for a being Element of NonZero F ex b being Element of NonZero F st
( a * b = 1. F & b * a = 1. F )
let a be Element of NonZero F; :: thesis: ex b being Element of NonZero F st
( a * b = 1. F & b * a = 1. F )
set B = H₁(F) \ {(0. F)};
set P = (omf F) ! H₁(F),(0. F);
A1: H₁(F) \ {(0. F)} = NonZero F ;
then reconsider e = 1. F as Element of H₁(F) \ {(0. F)} by STRUCT_0:2;
addLoopStr(# (H₁(F) \ {(0. F)}),((omf F) ! H₁(F),(0. F)),e #) is AbGroup by A1, Def11;
then consider D being strict AbGroup such that
A2: D = addLoopStr(# (H₁(F) \ {(0. F)}),((omf F) ! H₁(F),(0. F)),e #) ;
reconsider a = a as Element of D by A2;
consider b being Element of D such that
A3: a + b = 0. D and
A4: b + a = 0. D by Th7;
reconsider b = b as Element of NonZero F by A2;
take b ; :: thesis: ( a * b = 1. F & b * a = 1. F )
(omf F) || (H₁(F) \ {(0. F)}) is Function of [:(H₁(F) \ {(0. F)}),(H₁(F) \ {(0. F)}):],(H₁(F) \ {(0. F)}) by REALSET1:11;
then A5: dom ((omf F) || (H₁(F) \ {(0. F)})) = [:(H₁(F) \ {(0. F)}),(H₁(F) \ {(0. F)}):] by FUNCT_2:def 1;
for x, y being Element of H₁(F) \ {(0. F)} holds (omf F) . x,y = the addF of D . x,y

proof

let x, y be Element of H₁(F) \ {(0. F)}; :: thesis: (omf F) . x,y = the addF of D . x,y
[x,y] in [:(H₁(F) \ {(0. F)}),(H₁(F) \ {(0. F)}):] ;
hence (omf F) . x,y = the addF of D . x,y by A2, A5, FUNCT_1:70; :: thesis: verum

end;

hence ( a * b = 1. F & b * a = 1. F ) by A2, A3, A4; :: thesis: verum