:: Binary Operations on Numbers
:: by Library Committee
::
:: Received June 21, 2004
:: Copyright (c) 2004-2012 Association of Mizar Users


begin

scheme :: BINOP_2:sch 1
FuncDefUniq{ F1() ->  non empty set , F2() ->  non empty set , F3( Element of F1()) ->  set } :
 for f1, f2 being Function of F1(),F2() st ( for x being Element of F1() holds f1 . x = F3(x) ) & ( for x being Element of F1() holds f2 . x = F3(x) ) holds
f1 = f2
proofend;

scheme :: BINOP_2:sch 2
BinOpDefuniq{ F1() ->  non empty set , F2( Element of F1(), Element of F1()) ->  set } :
 for o1, o2 being BinOp of F1() st ( for a, b being Element of F1() holds o1 . (a,b) = F2(a,b) ) & ( for a, b being Element of F1() holds o2 . (a,b) = F2(a,b) ) holds
o1 = o2
proofend;

scheme :: BINOP_2:sch 3
CFuncDefUniq{ F1( complex number ) ->  set } :
 for f1, f2 being Function of COMPLEX,COMPLEX st ( for x being complex number holds f1 . x = F1(x) ) & ( for x being complex number holds f2 . x = F1(x) ) holds
f1 = f2
proofend;

scheme :: BINOP_2:sch 4
RFuncDefUniq{ F1( real number ) ->  set } :
 for f1, f2 being Function of REAL,REAL st ( for x being real number holds f1 . x = F1(x) ) & ( for x being real number holds f2 . x = F1(x) ) holds
f1 = f2
proofend;

registration
cluster  ->  rational for Element of RAT ;
coherence
 for b1 being Element of RAT holds b1 is rational by RAT_1:def_2;
end;

scheme :: BINOP_2:sch 5
WFuncDefUniq{ F1( rational number ) ->  set } :
 for f1, f2 being Function of RAT,RAT st ( for x being rational number holds f1 . x = F1(x) ) & ( for x being rational number holds f2 . x = F1(x) ) holds
f1 = f2
proofend;

scheme :: BINOP_2:sch 6
IFuncDefUniq{ F1( integer number ) ->  set } :
 for f1, f2 being Function of INT,INT st ( for x being integer number holds f1 . x = F1(x) ) & ( for x being integer number holds f2 . x = F1(x) ) holds
f1 = f2
proofend;

scheme :: BINOP_2:sch 7
NFuncDefUniq{ F1( Nat) ->  set } :
 for f1, f2 being Function of NAT,NAT st ( for x being Nat holds f1 . x = F1(x) ) & ( for x being Nat holds f2 . x = F1(x) ) holds
f1 = f2
proofend;

scheme :: BINOP_2:sch 8
CBinOpDefuniq{ F1( complex number , complex number ) ->  set } :
 for o1, o2 being BinOp of COMPLEX st ( for a, b being complex number holds o1 . (a,b) = F1(a,b) ) & ( for a, b being complex number holds o2 . (a,b) = F1(a,b) ) holds
o1 = o2
proofend;

scheme :: BINOP_2:sch 9
RBinOpDefuniq{ F1( real number , real number ) ->  set } :
 for o1, o2 being BinOp of REAL st ( for a, b being real number holds o1 . (a,b) = F1(a,b) ) & ( for a, b being real number holds o2 . (a,b) = F1(a,b) ) holds
o1 = o2
proofend;

scheme :: BINOP_2:sch 10
WBinOpDefuniq{ F1( rational number , rational number ) ->  set } :
 for o1, o2 being BinOp of RAT st ( for a, b being rational number holds o1 . (a,b) = F1(a,b) ) & ( for a, b being rational number holds o2 . (a,b) = F1(a,b) ) holds
o1 = o2
proofend;

scheme :: BINOP_2:sch 11
IBinOpDefuniq{ F1( integer number , integer number ) ->  set } :
 for o1, o2 being BinOp of INT st ( for a, b being integer number holds o1 . (a,b) = F1(a,b) ) & ( for a, b being integer number holds o2 . (a,b) = F1(a,b) ) holds
o1 = o2
proofend;

scheme :: BINOP_2:sch 12
NBinOpDefuniq{ F1( Nat, Nat) ->  set } :
 for o1, o2 being BinOp of NAT st ( for a, b being Nat holds o1 . (a,b) = F1(a,b) ) & ( for a, b being Nat holds o2 . (a,b) = F1(a,b) ) holds
o1 = o2
proofend;

scheme :: BINOP_2:sch 13
CLambda2D{ F1( complex number , complex number ) ->  complex number } :
 ex f being Function of [:COMPLEX,COMPLEX:],COMPLEX st
for x, y being complex number holds f . (x,y) = F1(x,y)
proofend;

scheme :: BINOP_2:sch 14
RLambda2D{ F1( real number , real number ) ->  real number } :
 ex f being Function of [:REAL,REAL:],REAL st
for x, y being real number holds f . (x,y) = F1(x,y)
proofend;

scheme :: BINOP_2:sch 15
WLambda2D{ F1( rational number , rational number ) ->  rational number } :
 ex f being Function of [:RAT,RAT:],RAT st
for x, y being rational number holds f . (x,y) = F1(x,y)
proofend;

scheme :: BINOP_2:sch 16
ILambda2D{ F1( integer number , integer number ) ->  integer number } :
 ex f being Function of [:INT,INT:],INT st
for x, y being integer number holds f . (x,y) = F1(x,y)
proofend;

scheme :: BINOP_2:sch 17
NLambda2D{ F1( Nat, Nat) ->  Nat } :
 ex f being Function of [:NAT,NAT:],NAT st
for x, y being Nat holds f . (x,y) = F1(x,y)
proofend;

scheme :: BINOP_2:sch 18
CLambdaD{ F1( complex number ) ->  complex number } :
 ex f being Function of COMPLEX,COMPLEX st
for x being complex number holds f . x = F1(x)
proofend;

scheme :: BINOP_2:sch 19
RLambdaD{ F1( real number ) ->  real number } :
 ex f being Function of REAL,REAL st
for x being real number holds f . x = F1(x)
proofend;

scheme :: BINOP_2:sch 20
WLambdaD{ F1( rational number ) ->  rational number } :
 ex f being Function of RAT,RAT st
for x being rational number holds f . x = F1(x)
proofend;

scheme :: BINOP_2:sch 21
ILambdaD{ F1( integer number ) ->  integer number } :
 ex f being Function of INT,INT st
for x being integer number holds f . x = F1(x)
proofend;

scheme :: BINOP_2:sch 22
NLambdaD{ F1( Nat) ->  Nat } :
 ex f being Function of NAT,NAT st
for x being Nat holds f . x = F1(x)
proofend;

definition
let c1 be complex number ;
:: original:-
redefine func - c1 ->  Element of COMPLEX ;
coherence
 - c1 is Element of COMPLEX by XCMPLX_0:def_2;
:: original:"
redefine funcc1 "  ->  Element of COMPLEX ;
coherence
 c1 " is Element of COMPLEX by XCMPLX_0:def_2;
let c2 be complex number ;
:: original:+
redefine funcc1 + c2 ->  Element of COMPLEX ;
coherence
 c1 + c2 is Element of COMPLEX by XCMPLX_0:def_2;
:: original:-
redefine funcc1 - c2 ->  Element of COMPLEX ;
coherence
 c1 - c2 is Element of COMPLEX by XCMPLX_0:def_2;
:: original:*
redefine funcc1 * c2 ->  Element of COMPLEX ;
coherence
 c1 * c2 is Element of COMPLEX by XCMPLX_0:def_2;
:: original:/
redefine funcc1 / c2 ->  Element of COMPLEX ;
coherence
 c1 / c2 is Element of COMPLEX by XCMPLX_0:def_2;
end;

definition
let r1 be real number ;
:: original:-
redefine func - r1 ->  Element of REAL ;
coherence
 - r1 is Element of REAL by XREAL_0:def_1;
:: original:"
redefine funcr1 "  ->  Element of REAL ;
coherence
 r1 " is Element of REAL by XREAL_0:def_1;
let r2 be real number ;
:: original:+
redefine funcr1 + r2 ->  Element of REAL ;
coherence
 r1 + r2 is Element of REAL by XREAL_0:def_1;
:: original:-
redefine funcr1 - r2 ->  Element of REAL ;
coherence
 r1 - r2 is Element of REAL by XREAL_0:def_1;
:: original:*
redefine funcr1 * r2 ->  Element of REAL ;
coherence
 r1 * r2 is Element of REAL by XREAL_0:def_1;
:: original:/
redefine funcr1 / r2 ->  Element of REAL ;
coherence
 r1 / r2 is Element of REAL by XREAL_0:def_1;
end;

definition
let w1 be rational number ;
:: original:-
redefine func - w1 ->  Element of RAT ;
coherence
 - w1 is Element of RAT by RAT_1:def_2;
:: original:"
redefine funcw1 "  ->  Element of RAT ;
coherence
 w1 " is Element of RAT by RAT_1:def_2;
let w2 be rational number ;
:: original:+
redefine funcw1 + w2 ->  Element of RAT ;
coherence
 w1 + w2 is Element of RAT by RAT_1:def_2;
:: original:-
redefine funcw1 - w2 ->  Element of RAT ;
coherence
 w1 - w2 is Element of RAT by RAT_1:def_2;
:: original:*
redefine funcw1 * w2 ->  Element of RAT ;
coherence
 w1 * w2 is Element of RAT by RAT_1:def_2;
:: original:/
redefine funcw1 / w2 ->  Element of RAT ;
coherence
 w1 / w2 is Element of RAT by RAT_1:def_2;
end;

definition
let i1 be integer number ;
:: original:-
redefine func - i1 ->  Element of INT ;
coherence
 - i1 is Element of INT by INT_1:def_2;
let i2 be integer number ;
:: original:+
redefine funci1 + i2 ->  Element of INT ;
coherence
 i1 + i2 is Element of INT by INT_1:def_2;
:: original:-
redefine funci1 - i2 ->  Element of INT ;
coherence
 i1 - i2 is Element of INT by INT_1:def_2;
:: original:*
redefine funci1 * i2 ->  Element of INT ;
coherence
 i1 * i2 is Element of INT by INT_1:def_2;
end;

definition
let n1, n2 be Nat;
:: original:+
redefine funcn1 + n2 ->  Element of NAT ;
coherence
 n1 + n2 is Element of NAT by ORDINAL1:def_12;
:: original:*
redefine funcn1 * n2 ->  Element of NAT ;
coherence
 n1 * n2 is Element of NAT by ORDINAL1:def_12;
end;

definition
func compcomplex  ->  UnOp of COMPLEX means :: BINOP_2:def 1
 for c being complex number holds it . c = - c;
existence
 ex b1 being UnOp of COMPLEX st
for c being complex number holds b1 . c = - c from BINOP_2:sch_18();
uniqueness
 for b1, b2 being UnOp of COMPLEX st ( for c being complex number holds b1 . c = - c ) & ( for c being complex number holds b2 . c = - c ) holds
b1 = b2 from BINOP_2:sch_3();
func invcomplex  ->  UnOp of COMPLEX means :: BINOP_2:def 2
 for c being complex number holds it . c = c " ;
existence
 ex b1 being UnOp of COMPLEX st
for c being complex number holds b1 . c = c " from BINOP_2:sch_18();
uniqueness
 for b1, b2 being UnOp of COMPLEX st ( for c being complex number holds b1 . c = c " ) & ( for c being complex number holds b2 . c = c " ) holds
b1 = b2 from BINOP_2:sch_3();
func addcomplex  ->  BinOp of COMPLEX means :Def3: :: BINOP_2:def 3
 for c1, c2 being complex number holds it . (c1,c2) = c1 + c2;
existence
 ex b1 being BinOp of COMPLEX st
for c1, c2 being complex number holds b1 . (c1,c2) = c1 + c2 from BINOP_2:sch_13();
uniqueness
 for b1, b2 being BinOp of COMPLEX st ( for c1, c2 being complex number holds b1 . (c1,c2) = c1 + c2 ) & ( for c1, c2 being complex number holds b2 . (c1,c2) = c1 + c2 ) holds
b1 = b2 from BINOP_2:sch_8();
func diffcomplex  ->  BinOp of COMPLEX means :: BINOP_2:def 4
 for c1, c2 being complex number holds it . (c1,c2) = c1 - c2;
existence
 ex b1 being BinOp of COMPLEX st
for c1, c2 being complex number holds b1 . (c1,c2) = c1 - c2 from BINOP_2:sch_13();
uniqueness
 for b1, b2 being BinOp of COMPLEX st ( for c1, c2 being complex number holds b1 . (c1,c2) = c1 - c2 ) & ( for c1, c2 being complex number holds b2 . (c1,c2) = c1 - c2 ) holds
b1 = b2 from BINOP_2:sch_8();
func multcomplex  ->  BinOp of COMPLEX means :Def5: :: BINOP_2:def 5
 for c1, c2 being complex number holds it . (c1,c2) = c1 * c2;
existence
 ex b1 being BinOp of COMPLEX st
for c1, c2 being complex number holds b1 . (c1,c2) = c1 * c2 from BINOP_2:sch_13();
uniqueness
 for b1, b2 being BinOp of COMPLEX st ( for c1, c2 being complex number holds b1 . (c1,c2) = c1 * c2 ) & ( for c1, c2 being complex number holds b2 . (c1,c2) = c1 * c2 ) holds
b1 = b2 from BINOP_2:sch_8();
func divcomplex  ->  BinOp of COMPLEX means :: BINOP_2:def 6
 for c1, c2 being complex number holds it . (c1,c2) = c1 / c2;
existence
 ex b1 being BinOp of COMPLEX st
for c1, c2 being complex number holds b1 . (c1,c2) = c1 / c2 from BINOP_2:sch_13();
uniqueness
 for b1, b2 being BinOp of COMPLEX st ( for c1, c2 being complex number holds b1 . (c1,c2) = c1 / c2 ) & ( for c1, c2 being complex number holds b2 . (c1,c2) = c1 / c2 ) holds
b1 = b2 from BINOP_2:sch_8();
end;

:: deftheorem  defines compcomplex BINOP_2:def_1_:_
 for b1 being UnOp of COMPLEX holds
( b1 = compcomplex iff for c being complex number holds b1 . c = - c );

:: deftheorem  defines invcomplex BINOP_2:def_2_:_
 for b1 being UnOp of COMPLEX holds
( b1 = invcomplex iff for c being complex number holds b1 . c = c " );

:: deftheorem Def3 defines addcomplex BINOP_2:def_3_:_
 for b1 being BinOp of COMPLEX holds
( b1 = addcomplex iff for c1, c2 being complex number holds b1 . (c1,c2) = c1 + c2 );

:: deftheorem  defines diffcomplex BINOP_2:def_4_:_
 for b1 being BinOp of COMPLEX holds
( b1 = diffcomplex iff for c1, c2 being complex number holds b1 . (c1,c2) = c1 - c2 );

:: deftheorem Def5 defines multcomplex BINOP_2:def_5_:_
 for b1 being BinOp of COMPLEX holds
( b1 = multcomplex iff for c1, c2 being complex number holds b1 . (c1,c2) = c1 * c2 );

:: deftheorem  defines divcomplex BINOP_2:def_6_:_
 for b1 being BinOp of COMPLEX holds
( b1 = divcomplex iff for c1, c2 being complex number holds b1 . (c1,c2) = c1 / c2 );

definition
func compreal  ->  UnOp of REAL means :: BINOP_2:def 7
 for r being real number holds it . r = - r;
existence
 ex b1 being UnOp of REAL st
for r being real number holds b1 . r = - r from BINOP_2:sch_19();
uniqueness
 for b1, b2 being UnOp of REAL st ( for r being real number holds b1 . r = - r ) & ( for r being real number holds b2 . r = - r ) holds
b1 = b2 from BINOP_2:sch_4();
func invreal  ->  UnOp of REAL means :: BINOP_2:def 8
 for r being real number holds it . r = r " ;
existence
 ex b1 being UnOp of REAL st
for r being real number holds b1 . r = r " from BINOP_2:sch_19();
uniqueness
 for b1, b2 being UnOp of REAL st ( for r being real number holds b1 . r = r " ) & ( for r being real number holds b2 . r = r " ) holds
b1 = b2 from BINOP_2:sch_4();
func addreal  ->  BinOp of REAL means :Def9: :: BINOP_2:def 9
 for r1, r2 being real number holds it . (r1,r2) = r1 + r2;
existence
 ex b1 being BinOp of REAL st
for r1, r2 being real number holds b1 . (r1,r2) = r1 + r2 from BINOP_2:sch_14();
uniqueness
 for b1, b2 being BinOp of REAL st ( for r1, r2 being real number holds b1 . (r1,r2) = r1 + r2 ) & ( for r1, r2 being real number holds b2 . (r1,r2) = r1 + r2 ) holds
b1 = b2 from BINOP_2:sch_9();
func diffreal  ->  BinOp of REAL means :: BINOP_2:def 10
 for r1, r2 being real number holds it . (r1,r2) = r1 - r2;
existence
 ex b1 being BinOp of REAL st
for r1, r2 being real number holds b1 . (r1,r2) = r1 - r2 from BINOP_2:sch_14();
uniqueness
 for b1, b2 being BinOp of REAL st ( for r1, r2 being real number holds b1 . (r1,r2) = r1 - r2 ) & ( for r1, r2 being real number holds b2 . (r1,r2) = r1 - r2 ) holds
b1 = b2 from BINOP_2:sch_9();
func multreal  ->  BinOp of REAL means :Def11: :: BINOP_2:def 11
 for r1, r2 being real number holds it . (r1,r2) = r1 * r2;
existence
 ex b1 being BinOp of REAL st
for r1, r2 being real number holds b1 . (r1,r2) = r1 * r2 from BINOP_2:sch_14();
uniqueness
 for b1, b2 being BinOp of REAL st ( for r1, r2 being real number holds b1 . (r1,r2) = r1 * r2 ) & ( for r1, r2 being real number holds b2 . (r1,r2) = r1 * r2 ) holds
b1 = b2 from BINOP_2:sch_9();
func divreal  ->  BinOp of REAL means :: BINOP_2:def 12
 for r1, r2 being real number holds it . (r1,r2) = r1 / r2;
existence
 ex b1 being BinOp of REAL st
for r1, r2 being real number holds b1 . (r1,r2) = r1 / r2 from BINOP_2:sch_14();
uniqueness
 for b1, b2 being BinOp of REAL st ( for r1, r2 being real number holds b1 . (r1,r2) = r1 / r2 ) & ( for r1, r2 being real number holds b2 . (r1,r2) = r1 / r2 ) holds
b1 = b2 from BINOP_2:sch_9();
end;

:: deftheorem  defines compreal BINOP_2:def_7_:_
 for b1 being UnOp of REAL holds
( b1 = compreal iff for r being real number holds b1 . r = - r );

:: deftheorem  defines invreal BINOP_2:def_8_:_
 for b1 being UnOp of REAL holds
( b1 = invreal iff for r being real number holds b1 . r = r " );

:: deftheorem Def9 defines addreal BINOP_2:def_9_:_
 for b1 being BinOp of REAL holds
( b1 = addreal iff for r1, r2 being real number holds b1 . (r1,r2) = r1 + r2 );

:: deftheorem  defines diffreal BINOP_2:def_10_:_
 for b1 being BinOp of REAL holds
( b1 = diffreal iff for r1, r2 being real number holds b1 . (r1,r2) = r1 - r2 );

:: deftheorem Def11 defines multreal BINOP_2:def_11_:_
 for b1 being BinOp of REAL holds
( b1 = multreal iff for r1, r2 being real number holds b1 . (r1,r2) = r1 * r2 );

:: deftheorem  defines divreal BINOP_2:def_12_:_
 for b1 being BinOp of REAL holds
( b1 = divreal iff for r1, r2 being real number holds b1 . (r1,r2) = r1 / r2 );

definition
func comprat  ->  UnOp of RAT means :: BINOP_2:def 13
 for w being rational number holds it . w = - w;
existence
 ex b1 being UnOp of RAT st
for w being rational number holds b1 . w = - w from BINOP_2:sch_20();
uniqueness
 for b1, b2 being UnOp of RAT st ( for w being rational number holds b1 . w = - w ) & ( for w being rational number holds b2 . w = - w ) holds
b1 = b2 from BINOP_2:sch_5();
func invrat  ->  UnOp of RAT means :: BINOP_2:def 14
 for w being rational number holds it . w = w " ;
existence
 ex b1 being UnOp of RAT st
for w being rational number holds b1 . w = w " from BINOP_2:sch_20();
uniqueness
 for b1, b2 being UnOp of RAT st ( for w being rational number holds b1 . w = w " ) & ( for w being rational number holds b2 . w = w " ) holds
b1 = b2 from BINOP_2:sch_5();
func addrat  ->  BinOp of RAT means :Def15: :: BINOP_2:def 15
 for w1, w2 being rational number holds it . (w1,w2) = w1 + w2;
existence
 ex b1 being BinOp of RAT st
for w1, w2 being rational number holds b1 . (w1,w2) = w1 + w2 from BINOP_2:sch_15();
uniqueness
 for b1, b2 being BinOp of RAT st ( for w1, w2 being rational number holds b1 . (w1,w2) = w1 + w2 ) & ( for w1, w2 being rational number holds b2 . (w1,w2) = w1 + w2 ) holds
b1 = b2 from BINOP_2:sch_10();
func diffrat  ->  BinOp of RAT means :: BINOP_2:def 16
 for w1, w2 being rational number holds it . (w1,w2) = w1 - w2;
existence
 ex b1 being BinOp of RAT st
for w1, w2 being rational number holds b1 . (w1,w2) = w1 - w2 from BINOP_2:sch_15();
uniqueness
 for b1, b2 being BinOp of RAT st ( for w1, w2 being rational number holds b1 . (w1,w2) = w1 - w2 ) & ( for w1, w2 being rational number holds b2 . (w1,w2) = w1 - w2 ) holds
b1 = b2 from BINOP_2:sch_10();
func multrat  ->  BinOp of RAT means :Def17: :: BINOP_2:def 17
 for w1, w2 being rational number holds it . (w1,w2) = w1 * w2;
existence
 ex b1 being BinOp of RAT st
for w1, w2 being rational number holds b1 . (w1,w2) = w1 * w2 from BINOP_2:sch_15();
uniqueness
 for b1, b2 being BinOp of RAT st ( for w1, w2 being rational number holds b1 . (w1,w2) = w1 * w2 ) & ( for w1, w2 being rational number holds b2 . (w1,w2) = w1 * w2 ) holds
b1 = b2 from BINOP_2:sch_10();
func divrat  ->  BinOp of RAT means :: BINOP_2:def 18
 for w1, w2 being rational number holds it . (w1,w2) = w1 / w2;
existence
 ex b1 being BinOp of RAT st
for w1, w2 being rational number holds b1 . (w1,w2) = w1 / w2 from BINOP_2:sch_15();
uniqueness
 for b1, b2 being BinOp of RAT st ( for w1, w2 being rational number holds b1 . (w1,w2) = w1 / w2 ) & ( for w1, w2 being rational number holds b2 . (w1,w2) = w1 / w2 ) holds
b1 = b2 from BINOP_2:sch_10();
end;

:: deftheorem  defines comprat BINOP_2:def_13_:_
 for b1 being UnOp of RAT holds
( b1 = comprat iff for w being rational number holds b1 . w = - w );

:: deftheorem  defines invrat BINOP_2:def_14_:_
 for b1 being UnOp of RAT holds
( b1 = invrat iff for w being rational number holds b1 . w = w " );

:: deftheorem Def15 defines addrat BINOP_2:def_15_:_
 for b1 being BinOp of RAT holds
( b1 = addrat iff for w1, w2 being rational number holds b1 . (w1,w2) = w1 + w2 );

:: deftheorem  defines diffrat BINOP_2:def_16_:_
 for b1 being BinOp of RAT holds
( b1 = diffrat iff for w1, w2 being rational number holds b1 . (w1,w2) = w1 - w2 );

:: deftheorem Def17 defines multrat BINOP_2:def_17_:_
 for b1 being BinOp of RAT holds
( b1 = multrat iff for w1, w2 being rational number holds b1 . (w1,w2) = w1 * w2 );

:: deftheorem  defines divrat BINOP_2:def_18_:_
 for b1 being BinOp of RAT holds
( b1 = divrat iff for w1, w2 being rational number holds b1 . (w1,w2) = w1 / w2 );

definition
func compint  ->  UnOp of INT means :: BINOP_2:def 19
 for i being integer number holds it . i = - i;
existence
 ex b1 being UnOp of INT st
for i being integer number holds b1 . i = - i from BINOP_2:sch_21();
uniqueness
 for b1, b2 being UnOp of INT st ( for i being integer number holds b1 . i = - i ) & ( for i being integer number holds b2 . i = - i ) holds
b1 = b2 from BINOP_2:sch_6();
func addint  ->  BinOp of INT means :Def20: :: BINOP_2:def 20
 for i1, i2 being integer number holds it . (i1,i2) = i1 + i2;
existence
 ex b1 being BinOp of INT st
for i1, i2 being integer number holds b1 . (i1,i2) = i1 + i2 from BINOP_2:sch_16();
uniqueness
 for b1, b2 being BinOp of INT st ( for i1, i2 being integer number holds b1 . (i1,i2) = i1 + i2 ) & ( for i1, i2 being integer number holds b2 . (i1,i2) = i1 + i2 ) holds
b1 = b2 from BINOP_2:sch_11();
func diffint  ->  BinOp of INT means :: BINOP_2:def 21
 for i1, i2 being integer number holds it . (i1,i2) = i1 - i2;
existence
 ex b1 being BinOp of INT st
for i1, i2 being integer number holds b1 . (i1,i2) = i1 - i2 from BINOP_2:sch_16();
uniqueness
 for b1, b2 being BinOp of INT st ( for i1, i2 being integer number holds b1 . (i1,i2) = i1 - i2 ) & ( for i1, i2 being integer number holds b2 . (i1,i2) = i1 - i2 ) holds
b1 = b2 from BINOP_2:sch_11();
func multint  ->  BinOp of INT means :Def22: :: BINOP_2:def 22
 for i1, i2 being integer number holds it . (i1,i2) = i1 * i2;
existence
 ex b1 being BinOp of INT st
for i1, i2 being integer number holds b1 . (i1,i2) = i1 * i2 from BINOP_2:sch_16();
uniqueness
 for b1, b2 being BinOp of INT st ( for i1, i2 being integer number holds b1 . (i1,i2) = i1 * i2 ) & ( for i1, i2 being integer number holds b2 . (i1,i2) = i1 * i2 ) holds
b1 = b2 from BINOP_2:sch_11();
end;

:: deftheorem  defines compint BINOP_2:def_19_:_
 for b1 being UnOp of INT holds
( b1 = compint iff for i being integer number holds b1 . i = - i );

:: deftheorem Def20 defines addint BINOP_2:def_20_:_
 for b1 being BinOp of INT holds
( b1 = addint iff for i1, i2 being integer number holds b1 . (i1,i2) = i1 + i2 );

:: deftheorem  defines diffint BINOP_2:def_21_:_
 for b1 being BinOp of INT holds
( b1 = diffint iff for i1, i2 being integer number holds b1 . (i1,i2) = i1 - i2 );

:: deftheorem Def22 defines multint BINOP_2:def_22_:_
 for b1 being BinOp of INT holds
( b1 = multint iff for i1, i2 being integer number holds b1 . (i1,i2) = i1 * i2 );

definition
func addnat  ->  BinOp of NAT means :Def23: :: BINOP_2:def 23
 for n1, n2 being Nat holds it . (n1,n2) = n1 + n2;
existence
 ex b1 being BinOp of NAT st
for n1, n2 being Nat holds b1 . (n1,n2) = n1 + n2 from BINOP_2:sch_17();
uniqueness
 for b1, b2 being BinOp of NAT st ( for n1, n2 being Nat holds b1 . (n1,n2) = n1 + n2 ) & ( for n1, n2 being Nat holds b2 . (n1,n2) = n1 + n2 ) holds
b1 = b2 from BINOP_2:sch_12();
func multnat  ->  BinOp of NAT means :Def24: :: BINOP_2:def 24
 for n1, n2 being Nat holds it . (n1,n2) = n1 * n2;
existence
 ex b1 being BinOp of NAT st
for n1, n2 being Nat holds b1 . (n1,n2) = n1 * n2 from BINOP_2:sch_17();
uniqueness
 for b1, b2 being BinOp of NAT st ( for n1, n2 being Nat holds b1 . (n1,n2) = n1 * n2 ) & ( for n1, n2 being Nat holds b2 . (n1,n2) = n1 * n2 ) holds
b1 = b2 from BINOP_2:sch_12();
end;

:: deftheorem Def23 defines addnat BINOP_2:def_23_:_
 for b1 being BinOp of NAT holds
( b1 = addnat iff for n1, n2 being Nat holds b1 . (n1,n2) = n1 + n2 );

:: deftheorem Def24 defines multnat BINOP_2:def_24_:_
 for b1 being BinOp of NAT holds
( b1 = multnat iff for n1, n2 being Nat holds b1 . (n1,n2) = n1 * n2 );

registration
cluster addcomplex  ->  commutative associative ;
coherence
 ( addcomplex is commutative & addcomplex is associative )
proofend;
cluster multcomplex  ->  commutative associative ;
coherence
 ( multcomplex is commutative & multcomplex is associative )
proofend;
cluster addreal  ->  commutative associative ;
coherence
 ( addreal is commutative & addreal is associative )
proofend;
cluster multreal  ->  commutative associative ;
coherence
 ( multreal is commutative & multreal is associative )
proofend;
cluster addrat  ->  commutative associative ;
coherence
 ( addrat is commutative & addrat is associative )
proofend;
cluster multrat  ->  commutative associative ;
coherence
 ( multrat is commutative & multrat is associative )
proofend;
cluster addint  ->  commutative associative ;
coherence
 ( addint is commutative & addint is associative )
proofend;
cluster multint  ->  commutative associative ;
coherence
 ( multint is commutative & multint is associative )
proofend;
cluster addnat  ->  commutative associative ;
coherence
 ( addnat is commutative & addnat is associative )
proofend;
cluster multnat  ->  commutative associative ;
coherence
 ( multnat is commutative & multnat is associative )
proofend;
end;

Lm1:  0 in NAT
;

then reconsider z = 0 as Element of COMPLEX by NUMBERS:20;

Lm2:  z is_a_unity_wrt addcomplex
proofend;

Lm3:  0 is_a_unity_wrt addreal
proofend;

reconsider z = 0 as Element of RAT by Lm1, NUMBERS:18;

Lm4:  z is_a_unity_wrt addrat
proofend;

reconsider z = 0 as Element of INT by Lm1, NUMBERS:17;

Lm5:  z is_a_unity_wrt addint
proofend;

Lm6:  0 is_a_unity_wrt addnat
proofend;

Lm7:  1 in NAT
;

then reconsider z = 1 as Element of COMPLEX by NUMBERS:20;

Lm8:  z is_a_unity_wrt multcomplex
proofend;

Lm9:  1 is_a_unity_wrt multreal
proofend;

reconsider z = 1 as Element of RAT by Lm7, NUMBERS:18;

Lm10:  z is_a_unity_wrt multrat
proofend;

reconsider z = 1 as Element of INT by Lm7, NUMBERS:17;

Lm11:  z is_a_unity_wrt multint
proofend;

Lm12:  1 is_a_unity_wrt multnat
proofend;

registration
cluster addcomplex  ->  having_a_unity ;
coherence
 addcomplex is having_a_unity by Lm2, SETWISEO:def_2;
cluster addreal  ->  having_a_unity ;
coherence
 addreal is having_a_unity by Lm3, SETWISEO:def_2;
cluster addrat  ->  having_a_unity ;
coherence
 addrat is having_a_unity by Lm4, SETWISEO:def_2;
cluster addint  ->  having_a_unity ;
coherence
 addint is having_a_unity by Lm5, SETWISEO:def_2;
cluster addnat  ->  having_a_unity ;
coherence
 addnat is having_a_unity by Lm6, SETWISEO:def_2;
cluster multcomplex  ->  having_a_unity ;
coherence
 multcomplex is having_a_unity by Lm8, SETWISEO:def_2;
cluster multreal  ->  having_a_unity ;
coherence
 multreal is having_a_unity by Lm9, SETWISEO:def_2;
cluster multrat  ->  having_a_unity ;
coherence
 multrat is having_a_unity by Lm10, SETWISEO:def_2;
cluster multint  ->  having_a_unity ;
coherence
 multint is having_a_unity by Lm11, SETWISEO:def_2;
cluster multnat  ->  having_a_unity ;
coherence
 multnat is having_a_unity by Lm12, SETWISEO:def_2;
end;

theorem :: BINOP_2:1
the_unity_wrt addcomplex = 0 by Lm2, BINOP_1:def_8;

theorem :: BINOP_2:2
the_unity_wrt addreal = 0 by Lm3, BINOP_1:def_8;

theorem :: BINOP_2:3
the_unity_wrt addrat = 0 by Lm4, BINOP_1:def_8;

theorem :: BINOP_2:4
the_unity_wrt addint = 0 by Lm5, BINOP_1:def_8;

theorem :: BINOP_2:5
the_unity_wrt addnat = 0 by Lm6, BINOP_1:def_8;

theorem :: BINOP_2:6
the_unity_wrt multcomplex = 1 by Lm8, BINOP_1:def_8;

theorem :: BINOP_2:7
the_unity_wrt multreal = 1 by Lm9, BINOP_1:def_8;

theorem :: BINOP_2:8
the_unity_wrt multrat = 1 by Lm10, BINOP_1:def_8;

theorem :: BINOP_2:9
the_unity_wrt multint = 1 by Lm11, BINOP_1:def_8;

theorem :: BINOP_2:10
the_unity_wrt multnat = 1 by Lm12, BINOP_1:def_8;