let S be non empty set ; :: thesis: for G being non empty right_complementable Abelian add-associative right_zeroed midpoint_operator addLoopStr
for w being Function of [:S,S:], the carrier of G st w is_atlas_of S,G holds
for a, b, c being Element of S holds
( (@ w) . (a,b) = c iff w . (a,c) = w . (c,b) )

let G be non empty right_complementable Abelian add-associative right_zeroed midpoint_operator addLoopStr ; :: thesis: for w being Function of [:S,S:], the carrier of G st w is_atlas_of S,G holds
for a, b, c being Element of S holds
( (@ w) . (a,b) = c iff w . (a,c) = w . (c,b) )

let w be Function of [:S,S:], the carrier of G; :: thesis: ( w is_atlas_of S,G implies for a, b, c being Element of S holds
( (@ w) . (a,b) = c iff w . (a,c) = w . (c,b) ) )

assume A1: w is_atlas_of S,G ; :: thesis: for a, b, c being Element of S holds
( (@ w) . (a,b) = c iff w . (a,c) = w . (c,b) )

let a, b, c be Element of S; :: thesis: ( (@ w) . (a,b) = c iff w . (a,c) = w . (c,b) )
thus ( (@ w) . (a,b) = c implies w . (a,c) = w . (c,b) ) by ; :: thesis: ( w . (a,c) = w . (c,b) implies (@ w) . (a,b) = c )
thus ( w . (a,c) = w . (c,b) implies (@ w) . (a,b) = c ) :: thesis: verum
proof
defpred S1[ Element of S, Element of S, Element of S] means w . (\$1,\$3) = w . (\$3,\$2);
assume A2: w . (a,c) = w . (c,b) ; :: thesis: (@ w) . (a,b) = c
A3: for a, b, c, c9 being Element of S st S1[a,b,c] & S1[a,b,c9] holds
c = c9
proof
let a, b, c, c9 be Element of S; :: thesis: ( S1[a,b,c] & S1[a,b,c9] implies c = c9 )
assume A4: ( S1[a,b,c] & S1[a,b,c9] ) ; :: thesis: c = c9
w . (c,c9) = (w . (c,a)) + (w . (a,c9)) by A1
.= (w . (c9,b)) + (w . (b,c)) by A1, A4, Th5
.= w . (c9,c) by A1
.= - (w . (c,c9)) by ;
then w . (c,c9) = 0. G by Th16;
hence c = c9 by ; :: thesis: verum
end;
set c9 = (@ w) . (a,b);
S1[a,b,(@ w) . (a,b)] by ;
hence (@ w) . (a,b) = c by A2, A3; :: thesis: verum
end;