Article REALSET3, MML version 4.99.1005
:: REALSET3:th 2
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr holds
(revf b1) . 1. b1 = 1. b1;
:: REALSET3:th 4
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2, b3 being Element of (the carrier of b1) \ {0. b1} holds
(revf b1) . ((omf b1) .(b2,(revf b1) . b3)) = (omf b1) .(b3,(revf b1) . b2);
:: REALSET3:th 6
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2, b3 being Element of (the carrier of b1) \ {0. b1} holds
(revf b1) . ((omf b1) .(b2,b3)) = (omf b1) .((revf b1) . b2,(revf b1) . b3);
:: REALSET3:th 7
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2, b3, b4, b5 being Element of the carrier of b1 holds
b2 - b3 = b4 - b5
iff
b2 + b5 = b3 + b4;
:: REALSET3:th 8
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2, b3 being Element of the carrier of b1
for b4, b5 being Element of (the carrier of b1) \ {0. b1} holds
(omf b1) .(b2,(revf b1) . b4) = (omf b1) .(b3,(revf b1) . b5)
iff
(omf b1) .(b2,b5) = (omf b1) .(b4,b3);
:: REALSET3:th 10
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2, b3 being Element of the carrier of b1
for b4, b5 being Element of (the carrier of b1) \ {0. b1} holds
(omf b1) .((omf b1) .(b2,(revf b1) . b4),(omf b1) .(b3,(revf b1) . b5)) = (omf b1) .((omf b1) .(b2,b3),(revf b1) . ((omf b1) .(b4,b5)));
:: REALSET3:th 11
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2, b3 being Element of the carrier of b1
for b4, b5 being Element of (the carrier of b1) \ {0. b1} holds
(the addF of b1) .((omf b1) .(b2,(revf b1) . b4),(omf b1) .(b3,(revf b1) . b5)) = (omf b1) .((the addF of b1) .((omf b1) .(b2,b5),(omf b1) .(b3,b4)),(revf b1) . ((omf b1) .(b4,b5)));
:: REALSET3:funcnot 1 => REALSET3:func 1
definition
let a1 be non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr;
func osf A1 -> Function-like quasi_total Relation of [:the carrier of a1,the carrier of a1:],the carrier of a1 means
for b1, b2 being Element of the carrier of a1 holds
it .(b1,b2) = (the addF of a1) .(b1,(comp a1) . b2);
end;
:: REALSET3:def 1
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2 being Function-like quasi_total Relation of [:the carrier of b1,the carrier of b1:],the carrier of b1 holds
b2 = osf b1
iff
for b3, b4 being Element of the carrier of b1 holds
b2 .(b3,b4) = (the addF of b1) .(b3,(comp b1) . b4);
:: REALSET3:th 14
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2 being Element of the carrier of b1 holds
(osf b1) .(b2,b2) = 0. b1;
:: REALSET3:th 15
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2, b3, b4 being Element of the carrier of b1 holds
(omf b1) .(b2,(osf b1) .(b3,b4)) = (osf b1) .((omf b1) .(b2,b3),(omf b1) .(b2,b4));
:: REALSET3:th 17
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2, b3, b4 being Element of the carrier of b1 holds
(omf b1) .((osf b1) .(b2,b3),b4) = (osf b1) .((omf b1) .(b2,b4),(omf b1) .(b3,b4));
:: REALSET3:th 18
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2, b3 being Element of the carrier of b1 holds
(osf b1) .(b2,b3) = (comp b1) . ((osf b1) .(b3,b2));
:: REALSET3:th 19
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2, b3 being Element of the carrier of b1 holds
(osf b1) .((comp b1) . b2,b3) = (comp b1) . ((the addF of b1) .(b2,b3));
:: REALSET3:th 20
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2, b3, b4, b5 being Element of the carrier of b1 holds
(osf b1) .(b2,b3) = (osf b1) .(b4,b5)
iff
b2 + b5 = b3 + b4;
:: REALSET3:th 21
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2 being Element of the carrier of b1 holds
(osf b1) .(0. b1,b2) = (comp b1) . b2;
:: REALSET3:th 22
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2 being Element of the carrier of b1 holds
(osf b1) .(b2,0. b1) = b2;
:: REALSET3:th 23
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2, b3, b4 being Element of the carrier of b1 holds
b2 + b3 = b4
iff
(osf b1) .(b4,b2) = b3;
:: REALSET3:th 24
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2, b3, b4 being Element of the carrier of b1 holds
b2 + b3 = b4
iff
(osf b1) .(b4,b3) = b2;
:: REALSET3:th 25
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2, b3, b4 being Element of the carrier of b1 holds
(osf b1) .(b2,(osf b1) .(b3,b4)) = (the addF of b1) .((osf b1) .(b2,b3),b4);
:: REALSET3:th 26
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2, b3, b4 being Element of the carrier of b1 holds
(osf b1) .(b2,(the addF of b1) .(b3,b4)) = (osf b1) .((osf b1) .(b2,b3),b4);
:: REALSET3:funcnot 2 => REALSET3:func 2
definition
let a1 be non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr;
func ovf A1 -> Function-like quasi_total Relation of [:the carrier of a1,(the carrier of a1) \ {0. a1}:],the carrier of a1 means
for b1 being Element of the carrier of a1
for b2 being Element of (the carrier of a1) \ {0. a1} holds
it .(b1,b2) = (omf a1) .(b1,(revf a1) . b2);
end;
:: REALSET3:def 2
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2 being Function-like quasi_total Relation of [:the carrier of b1,(the carrier of b1) \ {0. b1}:],the carrier of b1 holds
b2 = ovf b1
iff
for b3 being Element of the carrier of b1
for b4 being Element of (the carrier of b1) \ {0. b1} holds
b2 .(b3,b4) = (omf b1) .(b3,(revf b1) . b4);
:: REALSET3:th 29
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2 being Element of (the carrier of b1) \ {0. b1} holds
(ovf b1) .(b2,b2) = 1. b1;
:: REALSET3:th 31
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2, b3 being Element of the carrier of b1
for b4 being Element of (the carrier of b1) \ {0. b1} holds
(omf b1) .(b2,(ovf b1) .(b3,b4)) = (ovf b1) .((omf b1) .(b2,b3),b4);
:: REALSET3:th 32
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2 being Element of (the carrier of b1) \ {0. b1} holds
(omf b1) .(b2,(ovf b1) .(1. b1,b2)) = 1. b1 &
(omf b1) .((ovf b1) .(1. b1,b2),b2) = 1. b1;
:: REALSET3:th 35
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2, b3 being Element of (the carrier of b1) \ {0. b1} holds
(ovf b1) .(b2,b3) = (revf b1) . ((ovf b1) .(b3,b2));
:: REALSET3:th 36
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2, b3 being Element of (the carrier of b1) \ {0. b1} holds
(ovf b1) .((revf b1) . b2,b3) = (revf b1) . ((omf b1) .(b2,b3));
:: REALSET3:th 37
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2, b3 being Element of the carrier of b1
for b4, b5 being Element of (the carrier of b1) \ {0. b1} holds
(ovf b1) .(b2,b4) = (ovf b1) .(b3,b5)
iff
(omf b1) .(b2,b5) = (omf b1) .(b4,b3);
:: REALSET3:th 38
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2 being Element of (the carrier of b1) \ {0. b1} holds
(ovf b1) .(1. b1,b2) = (revf b1) . b2;
:: REALSET3:th 39
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2 being Element of the carrier of b1 holds
(ovf b1) .(b2,1. b1) = b2;
:: REALSET3:th 40
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2 being Element of (the carrier of b1) \ {0. b1}
for b3, b4 being Element of the carrier of b1 holds
(omf b1) .(b2,b3) = b4
iff
(ovf b1) .(b4,b2) = b3;
:: REALSET3:th 41
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2, b3 being Element of the carrier of b1
for b4 being Element of (the carrier of b1) \ {0. b1} holds
(omf b1) .(b2,b4) = b3
iff
(ovf b1) .(b3,b4) = b2;
:: REALSET3:th 42
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2 being Element of the carrier of b1
for b3, b4 being Element of (the carrier of b1) \ {0. b1} holds
(ovf b1) .(b2,(ovf b1) .(b3,b4)) = (omf b1) .((ovf b1) .(b2,b3),b4);
:: REALSET3:th 43
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2 being Element of the carrier of b1
for b3, b4 being Element of (the carrier of b1) \ {0. b1} holds
(ovf b1) .(b2,(omf b1) .(b3,b4)) = (ovf b1) .((ovf b1) .(b2,b3),b4);