let a3, b3 be Real; :: thesis: ( ex a1, a2 being Real st
( (a1 + a2) + a3 = 1 & p = ((a1 * p1) + (a2 * p2)) + (a3 * p3) ) & ex a1, a2 being Real st
( (a1 + a2) + b3 = 1 & p = ((a1 * p1) + (a2 * p2)) + (b3 * p3) ) implies a3 = b3 )
assume A3: ( ex a1, a2 being Real st
( (a1 + a2) + a3 = 1 & p = ((a1 * p1) + (a2 * p2)) + (a3 * p3) ) & ex a1, a2 being Real st
( (a1 + a2) + b3 = 1 & p = ((a1 * p1) + (a2 * p2)) + (b3 * p3) ) ) ; :: thesis: a3 = b3
then consider a01, a02 being Real such that
A4: ( (a01 + a02) + a3 = 1 & p = ((a01 * p1) + (a02 * p2)) + (a3 * p3) ) ;
consider a1, a2 being Real such that
A5: ( (a1 + a2) + b3 = 1 & p = ((a1 * p1) + (a2 * p2)) + (b3 * p3) ) by A3;
consider a001, a002, a003 being Real such that
A6: ( p = ((a001 * p1) + (a002 * p2)) + (a003 * p3) & (a001 + a002) + a003 = 1 & ( for b01, b02, b03 being Real st p = ((b01 * p1) + (b02 * p2)) + (b03 * p3) & (b01 + b02) + b03 = 1 holds
( b01 = a001 & b02 = a002 & b03 = a003 ) ) ) by A1, Th60;
( a01 = a001 & a02 = a002 & a3 = a003 ) by A4, A6;
hence a3 = b3 by A5, A6; :: thesis: verum