set G = multMagma(# REAL,addreal #);
thus for h, g, f being Element of multMagma(# REAL,addreal #) holds (h * g) * f = h * (g * f) :: according to GROUP_1:def 3 :: thesis: multMagma(# REAL,addreal #) is Group-like
proof
let h, g, f be Element of multMagma(# REAL,addreal #); :: thesis: (h * g) * f = h * (g * f)
reconsider A = h, B = g, C = f as Real ;
A1: g * f = B + C by BINOP_2:def 9;
h * g = A + B by BINOP_2:def 9;
hence (h * g) * f = (A + B) + C by BINOP_2:def 9
.= A + (B + C)
.= h * (g * f) by A1, BINOP_2:def 9 ;
:: thesis: verum
end;
reconsider e = 0 as Element of multMagma(# REAL,addreal #) by XREAL_0:def 1;
take e ; :: according to GROUP_1:def 2 :: thesis: for h being Element of multMagma(# REAL,addreal #) holds
( h * e = h & e * h = h & ex g being Element of multMagma(# REAL,addreal #) st
( h * g = e & g * h = e ) )
let h be Element of multMagma(# REAL,addreal #); :: thesis: ( h * e = h & e * h = h & ex g being Element of multMagma(# REAL,addreal #) st
( h * g = e & g * h = e ) )
reconsider A = h as Real ;
thus h * e = A + 0 by BINOP_2:def 9
.= h ; :: thesis: ( e * h = h & ex g being Element of multMagma(# REAL,addreal #) st
( h * g = e & g * h = e ) )
thus e * h = 0 + A by BINOP_2:def 9
.= h ; :: thesis: ex g being Element of multMagma(# REAL,addreal #) st
( h * g = e & g * h = e )
reconsider g = - A as Element of multMagma(# REAL,addreal #) by XREAL_0:def 1;
take g ; :: thesis: ( h * g = e & g * h = e )
thus h * g = A + (- A) by BINOP_2:def 9
.= e ; :: thesis: g * h = e
thus g * h = (- A) + A by BINOP_2:def 9
.= e ; :: thesis: verum